On the regular counting function applied to the highly composite numbers: a(n) = A010846(A002182(n)). Michael De Vlieger, St. Louis, Missouri, 201803281030, revised 201803291700. Concerns sequences: A000005: Divisor counting function tau(n), i.e., number of numbers 1 <= d <= n such that d | n^e with 0 <= e <= 1. A000079: Nonnegative integer powers of 2. A001221: omega(n) = Number of distinct prime divisors of n. A002110: The primorials. A002182: Highly composite numbers, i.e., where records are set in A000005. A002183: Records in A000005 = A000005(A002182(n)). A010846: Regular counting function, abbreviated rcf(n), i.e., number of numbers 1 <= m <= n such that m | n^e with e >= 0. A054841: For n = Product_p^e, write e_k in the k-th place, or write 0 if p_k does not divide n. A060735: Irregular triangle read by rows: T(n, k) = k * A002110(n) for 1 <= k < prime(n + 1). A108602: A001221(A002182(n)). A244052: Highly regular numbers, i.e., where records are set in A010846. A288813: Irregular triangle read by rows: T(m, k) is the list of squarefree numbers A002110(m) < t < 2*A002110(m) such that A001221(t) = m. A301892: a(n) = A010846(A002182(n)). A301893: Where records occur in A010846(n)/A000005(n). We define an "n-regular" number as 1 <= m <= n such that m | n^e with integer e >= 0. A number that is not regular is said to be "non-regular". The divisor d is a special case of regular number m such that d | n^e with e = 0 or e = 1. Regular numbers m can exceed n; we are concerned only with regulars m <= n herein. Regular numbers, apart from 1, occupy the cototient of n. The union of the regulars and the cototient leaves us with the "semitotative" for composite n > 6 listed in row n of A272619. We are not concerned with "semidivisors" m such that m | n^e with e > 1, listed in row n of A272618, but with the regular and divisor counting functions A010846(n) and A000005(n), respectively. +-----------+ | Contents. | +-----------+ Table 1: Comparison of A010846(A002182(n)) and A002183(n). Table 2: A301893(i) = Numbers k that set records for the ratio rcf(k)/tau(k). Table 3: The intersection of A002182 and A244052. Table 4: Plot {omega(m)/A002110(omega(m))} for m in A002182. Graph 5: Plot "x" at {A108602(m), m/A002110(A108602(m))} for m in A002182. Table 6: m/A002110(A108602(m)) for m in A002182. (Observations and conjectures follow each table.) Code. +------------------------------------------------------------+ | Table 1: Comparison of A010846(A002182(n)) and A002183(n). | +------------------------------------------------------------+ Questions: How many regular numbers do the highly composite numbers have, given that the divisor is a special case of regular number? Do they also set records for the regular counting function (i.e., are the highly composite numbers also "highly regular")? Are the HCNs representative of numbers that set records for the ratio rcf(n)/tau(n)? (1) Position of A002182(n) in A244052. If A002182(n) is not in A244052, left blank. (2) Position of A002182(n) in b(n) = A301893 (see Table 2). (A301893 computed through 36,000,000). b(n) = numbers k that set records for the ratio rcf(k)/tau(k). A108602(n) = A001221(A002182(n)). a(n) = A301892(n) = A010846(A002182(n)). (3) "Multiplicity Family" of m = A002182(n): MN( m/A002110(A108602(m)) ). (See Table 4). MN = Multiplicity Notation = A054841(a(n)) here represents the reverse of A054841 in OEIS; i.e., "little-endian" rather than big-endian, e.g., A054841(84) = 1013 but here it is 2101. Since all terms of a(n) are squarefree, all digits in this number are either 0 or 1. We use A054841 or "multiplicity notation" here since the terms of a(n) are squarefree products of relatively small primes and the notation succinctly expresses their prime decomposition. This is notation akin to that of Achim Flammenkamp's studies of highly composite numbers. n A002182(n) (1) (2) A002183(n) A301892(n) A002183(n)/a(n) A108602(n) (3) ------------------------------------------------------------------------------------------------------ 1 1 1 1 1 1 1. 0 0 2 2 2 2 2 1. 1 0 3 4 3 3 3 1. 1 1 4 6 4 2 4 5 0.8 2 0 5 12 6 6 8 0.75 2 1 6 24 8 8 11 0.727273 2 2 7 36 9 14 0.642857 2 11 8 48 10 15 0.666667 2 3 9 60 11 12 26 0.461538 3 1 10 120 14 16 36 0.444444 3 2 11 180 16 18 44 0.409091 3 11 12 240 20 49 0.408163 3 3 13 360 24 58 0.413793 3 21 14 720 30 76 0.394737 3 31 15 840 22 32 131 0.244275 4 2 16 1260 24 36 156 0.230769 4 11 17 1680 26 40 174 0.229885 4 3 18 2520 48 206 0.23301 4 21 19 5040 60 266 0.225564 4 31 20 7560 64 308 0.207792 4 22 21 10080 72 339 0.212389 4 41 22 15120 80 388 0.206186 4 32 23 20160 84 428 0.196262 4 51 24 25200 90 460 0.195652 4 311 25 27720 47 96 766 0.125326 5 21 26 45360 100 550 0.181818 4 33 27 50400 108 568 0.190141 4 411 28 55440 120 979 0.122574 5 31 29 83160 128 1124 0.113879 5 22 30 110880 144 1238 0.116317 5 41 31 166320 160 1411 0.113395 5 32 32 221760 168 1548 0.108527 5 51 33 277200 180 1659 0.108499 5 311 34 332640 192 1754 0.109464 5 42 35 498960 200 1983 0.100857 5 33 36 554400 216 2048 0.105469 5 411 37 665280 224 2160 0.103704 5 52 38 720720 240 3689 0.0650583 6 31 39 1081080 256 4211 0.0607932 6 22 40 1441440 288 4617 0.0623782 6 41 41 2162160 320 5245 0.0610105 6 32 42 2882880 336 5731 0.0586285 6 51 43 3603600 360 6135 0.0586797 6 311 44 4324320 384 6482 0.059241 6 42 45 6486480 400 7308 0.0547345 6 33 46 7207200 432 7539 0.057302 6 411 47 8648640 448 7949 0.0563593 6 52 48 10810800 480 8477 0.0566238 6 321 49 14414400 504 9198 0.0547945 6 511 50 17297280 512 9681 0.0528871 6 62 51 21621600 576 10306 0.0558898 6 421 52 32432400 600 11515 0.0521059 6 331 53 36756720 - 640 19994 0.0320096 7 32 54 43243200 - 672 12448 0.0539846 6 521 55 61261200 - 720 23259 0.0309558 7 311 56 73513440 - 768 24532 0.031306 7 42 57 110270160 - 800 27570 0.029017 7 33 58 122522400 - 864 28408 0.030414 7 411 59 147026880 - 896 29917 0.0299495 7 52 60 183783600 - 960 31849 0.0301422 7 321 61 245044800 - 1008 34500 0.0292174 7 511 62 294053760 - 1024 36278 0.0282265 7 62 63 367567200 - 1152 38557 0.0298778 7 421 64 551350800 - 1200 43016 0.0278966 7 331 65 698377680 - 1280 77028 0.0166173 8 32 66 735134400 - 1344 46436 0.0289431 7 521 67 1102701600 - 1440 51652 0.0278789 7 431 68 1396755360 - 1536 93907 0.0163566 8 42 69 2095133040 - 1600 105210 0.0152077 8 33 70 2205403200 - 1680 61730 0.0272153 7 531 71 2327925600 - 1728 108330 0.0159513 8 411 72 2793510720 - 1792 113928 0.0157292 8 52 73 3491888400 - 1920 121118 0.0158523 8 321 74 4655851200 - 2016 130970 0.0153928 8 511 75 5587021440 - 2048 137566 0.0148874 8 62 76 6983776800 - 2304 146038 0.0157767 8 421 77 10475665200 - 2400 162599 0.0147602 8 331 78 13967553600 - 2688 175323 0.0153317 8 521 79 20951330400 - 2880 194715 0.0147908 8 431 80 27935107200 - 3072 209588 0.0146573 8 621 81 41902660800 - 3360 232236 0.014468 8 531 82 48886437600 - 3456 241387 0.0143173 8 4211 83 64250746560 - 3584 440351 0.00813896 9 52 84 73329656400 - 3600 266983 0.013484 8 3311 85 80313433200 - 3840 467403 0.00821561 9 321 86 97772875200 - 4032 286547 0.014071 8 5211 87 128501493120 - 4096 529244 0.00773934 9 62 88 146659312800 - 4320 316236 0.0136607 8 4311 89 160626866400 - 4608 561065 0.00821295 9 421 90 240940299600 - 4800 623240 0.00770169 9 331 91 293318625600 - 5040 373244 0.0135032 8 5311 92 321253732800 - 5376 670993 0.00801201 9 521 93 481880599200 - 5760 743774 0.00774429 9 431 94 642507465600 - 6144 799574 0.00768409 9 621 95 963761198400 - 6720 884524 0.00759731 9 531 96 1124388064800 - 6912 918857 0.00752239 9 4211 97 1606268664000 - 7168 1002878 0.00714743 9 522 98 1686582097200 - 7200 1014877 0.00709446 9 3311 99 1927522396800 - 7680 1048383 0.00732557 9 631 100 2248776129600 - 8064 1088291 0.00740978 9 5211 101 3212537328000 - 8192 1185830 0.00690824 9 622 102 3373164194400 - 8640 1199750 0.0072015 9 4311 103 4497552259200 - 9216 1284854 0.0071728 9 6211 104 6746328388800 - 10080 1413882 0.00712931 9 5311 105 8995104518400 - 10368 1512270 0.00685592 9 7211 106 9316358251200 - 10752 2607576 0.00412337 10 521 107 13492656777600 - 11520 1661238 0.00693459 9 6311 108 18632716502400 - 12288 3095217 0.00397 10 621 109 26985313555200 - 12960 1946228 0.00665903 9 7311 110 27949074753600 - 13440 3417004 0.00393327 10 531 111 32607253879200 - 13824 3547008 0.00389737 10 4211 112 46581791256000 - 14336 3864992 0.00370919 10 522 113 48910880818800 - 14400 3910396 0.00368249 10 3311 114 55898149507200 - 15360 4037208 0.00380461 10 631 115 65214507758400 - 16128 4188170 0.00385085 10 5211 116 93163582512000 - 16384 4557034 0.00359532 10 622 117 97821761637600 - 17280 4609691 0.00374862 10 4311 118 130429015516800 - 18432 4931474 0.00373762 10 6211 119 195643523275200 - 20160 5419232 0.00372008 10 5311 120 260858031033600 - 20736 5791113 0.00358066 10 7211 121 288807105787200 - 21504 10174493 0.00211352 11 521 122 391287046550400 - 23040 6354166 0.00362597 10 6311 123 577614211574400 - 24576 12036043 0.00204187 11 621 124 782574093100800 - 25920 7431427 0.00348789 10 7311 125 866421317361600 - 26880 13263098 0.00202668 11 531 126 1010824870255200 - 27648 13758600 0.00200951 11 4211 127 1444035528936000 - 28672 14970020 0.00191529 11 522 128 1516237305382800 - 28800 15143047 0.00190186 11 3311 129 1732842634723200 - 30720 15625881 0.00196597 11 631 130 2021649740510400 - 32256 16200656 0.00199103 11 5211 131 2888071057872000 - 32768 17604683 0.00186132 11 622 132 3032474610765600 - 34560 17805134 0.00194101 11 4311 133 4043299481020800 - 36864 19029413 0.00193721 11 6211 134 6064949221531200 - 40320 20884747 0.0019306 11 5311 135 8086598962041600 - 41472 22298971 0.00185982 11 7211 ////// Observations ////// (These observations do not include those well-known by those interested in HCNs) 1. The ratio tau(x)/rcf(x) for x = A002182(n) generally declines as x increases. This implies that for numbers that set records in the divisor counting function, the number of regular numbers is very much higher as n increases. 2. Only 13 HCNs are also "highly regular", i.e., appear in A244052. The largest HCN that is also highly regular is 27720, the 25th HCN and the 47th highly regular number. This seems to be associated with the fact that the prime decomposition of numbers in A002182 involves "more enriched" small primes as n increases, while the numbers in A244052 are either squarefree or in A060735. 3. Only 2 and 6 set records for the ratio tau(x)/rcf(x). In other words, the highly composite numbers seem generally not representative of the highest possible ratio of number of divisors to the number of regulars. 4. The value of A301892(n) jumps when omega(n) increases, in line with what we would expect of the regular counting function regarding not-necessarily squarefree products of n versus (n + 1) smallest primes. ////// Conjectures /////// 1. The intersection of A002182 and A244052 is finite, consisting of 13 terms: {1, 2, 4, 6, 12, 24, 60, 120, 180, 840, 1260, 1680, 27720}. All of these terms are also in A060735 and not in A288813, as the latter are squarefree and have "gaps" among prime divisors. This intersection has the following number of terms in the "tiers" 0 through 5 of A244052: {1, 2, 3, 3, 3, 1}. A "tier" n consists of all terms m in A244052 with A002110(n) <= m < A002110(n + 1). If we look at A060735 as a number triangle T(n,k) = k * A002110(n) with 1 <= k < prime(n + 1), the terms are: {0, 1}, {{1,1}, {1,2}}, {{2,1}, {2,2}, {2,4}}, {{3,2}, {3,4}, {3,6}}, {{4,4}, {4,6}, {4,8}}, {5,12} (It is this observation that spurred development of Tables 3, 4, 6, and Graph 5.) 2. If it is true that the only non-squarefree terms in b(n) are {18, 150}, then the intersection of A002182 and b(n) is finite and consists of {1, 6}. (see Table 2 Conjecture 1). +-------------------------------------------------------------------------------+ | Table 2: A301893(i) = Numbers k that set records for the ratio rcf(k)/tau(k). | +-------------------------------------------------------------------------------+ Question: What numbers set records for the ratio of the regular counting function and the divisor counting function? (1) Position of b(n) in A244052(n). If b(n) is not in A244052(n), left blank. PC(n) = A287352(n) is the "pi-code" or first differences of indices of prime divisors p of n, e.g., A287352(60) = 1,0,1,1 since 60 = 2 * 2 * 3 * 5. The terms are delimited by (.). i k = b(i) (1) rcf(k) tau(k) tau(k)/rcf(k) PC(k). ----------------------------------------------------------------------------------- 1 1 1 1 1 1. 0 2 6 4 5 4 0.8 1.1 3 10 5 6 4 0.666667 1.2 4 18 7 10 6 0.6 1.1.0 5 22 7 4 0.571429 1.4 6 30 9 18 8 0.444444 1.1.1 7 42 10 19 8 0.421053 1.1.2 8 66 22 8 0.363636 1.1.3 9 78 23 8 0.347826 1.1.4 10 102 25 8 0.32 1.1.5 11 114 26 8 0.307692 1.1.6 12 138 27 8 0.296296 1.1.7 13 150 15 41 12 0.292683 1.1.1.0 14 174 29 8 0.275862 1.1.8 15 210 17 68 16 0.235294 1.1.1.1 16 330 18 77 16 0.207792 1.1.1.2 17 390 19 80 16 0.2 1.1.1.3 18 510 86 16 0.186047 1.1.1.4 19 570 88 16 0.181818 1.1.1.5 20 690 94 16 0.170213 1.1.1.6 21 870 101 16 0.158416 1.1.1.7 22 1110 106 16 0.150943 1.1.1.9 23 1230 110 16 0.145455 1.1.1.10 24 1290 112 16 0.142857 1.1.1.11 25 1410 114 16 0.140351 1.1.1.12 26 1590 118 16 0.135593 1.1.1.13 27 1770 121 16 0.132231 1.1.1.14 28 1830 122 16 0.131148 1.1.1.15 29 2010 126 16 0.126984 1.1.1.16 30 2130 128 16 0.125 1.1.1.17 31 2190 130 16 0.123077 1.1.1.18 32 2310 29 283 32 0.113074 1.1.1.1.1 33 2730 30 295 32 0.108475 1.1.1.1.2 34 3570 31 313 32 0.102236 1.1.1.1.3 35 3990 32 322 32 0.0993789 1.1.1.1.4 36 4830 339 32 0.0943953 1.1.1.1.5 37 6090 359 32 0.0891365 1.1.1.1.6 38 6510 366 32 0.0874317 1.1.1.1.7 39 7770 382 32 0.0837696 1.1.1.1.8 40 8610 394 32 0.0812183 1.1.1.1.9 41 9030 399 32 0.0802005 1.1.1.1.10 42 9870 409 32 0.0782396 1.1.1.1.11 43 11130 422 32 0.0758294 1.1.1.1.12 44 12390 436 32 0.0733945 1.1.1.1.13 45 12810 440 32 0.0727273 1.1.1.1.14 46 14070 450 32 0.0711111 1.1.1.1.15 47 14910 456 32 0.0701754 1.1.1.1.16 48 15330 459 32 0.0697168 1.1.1.1.17 49 16590 471 32 0.0679406 1.1.1.1.18 50 17430 477 32 0.067086 1.1.1.1.19 51 18690 487 32 0.0657084 1.1.1.1.20 52 20370 499 32 0.0641283 1.1.1.1.21 53 21210 505 32 0.0633663 1.1.1.1.22 54 21630 508 32 0.0629921 1.1.1.1.23 55 22470 512 32 0.0625 1.1.1.1.24 56 22890 514 32 0.0622568 1.1.1.1.25 57 23730 518 32 0.0617761 1.1.1.1.26 58 26670 536 32 0.0597015 1.1.1.1.27 59 27510 540 32 0.0592593 1.1.1.1.28 60 28770 547 32 0.0585009 1.1.1.1.29 61 29190 550 32 0.0581818 1.1.1.1.30 62 30030 48 1161 64 0.0551249 1.1.1.1.1.1 63 36330 584 32 0.0547945 1.1.1.1.36 64 37590 589 32 0.0543294 1.1.1.1.37 65 38010 591 32 0.0541455 1.1.1.1.38 66 39270 49 1224 64 0.0522876 1.1.1.1.1.2 67 43890 50 1253 64 0.0510774 1.1.1.1.1.3 68 46410 51 1257 64 0.0509149 1.1.1.1.2.1 69 48930 630 32 0.0507937 1.1.1.1.47 70 50190 635 32 0.0503937 1.1.1.1.48 71 50610 637 32 0.0502355 1.1.1.1.49 72 51870 52 1285 64 0.0498054 1.1.1.1.2.2 73 52710 644 32 0.0496894 1.1.1.1.50 74 53130 53 1306 64 0.0490046 1.1.1.1.1.4 75 56490 656 32 0.0487805 1.1.1.1.53 76 56910 657 32 0.0487062 1.1.1.1.54 77 58170 660 32 0.0484848 1.1.1.1.55 78 59010 662 32 0.0483384 1.1.1.1.56 79 59430 663 32 0.0482655 1.1.1.1.57 80 61530 671 32 0.04769 1.1.1.1.58 81 64470 678 32 0.0471976 1.1.1.1.59 82 65310 681 32 0.0469897 1.1.1.1.60 83 65730 684 32 0.0467836 1.1.1.1.61 84 66570 686 32 0.0466472 1.1.1.1.62 85 66990 1376 64 0.0465116 1.1.1.1.1.5 86 69510 692 32 0.0462428 1.1.1.1.63 87 70770 695 32 0.0460432 1.1.1.1.64 88 71610 1397 64 0.0458125 1.1.1.1.1.6 89 72870 701 32 0.0456491 1.1.1.1.65 90 73290 702 32 0.045584 1.1.1.1.66 91 74130 705 32 0.0453901 1.1.1.1.67 92 75390 707 32 0.0452617 1.1.1.1.68 93 77070 711 32 0.045007 1.1.1.1.69 94 78330 714 32 0.0448179 1.1.1.1.70 95 79590 718 32 0.0445682 1.1.1.1.71 96 80430 719 32 0.0445063 1.1.1.1.72 97 81690 722 32 0.0443213 1.1.1.1.73 98 83370 726 32 0.0440771 1.1.1.1.74 99 84210 728 32 0.043956 1.1.1.1.75 100 85470 1457 64 0.0439259 1.1.1.1.1.7 101 85890 732 32 0.0437158 1.1.1.1.76 102 87990 738 32 0.0433604 1.1.1.1.77 103 88410 739 32 0.0433018 1.1.1.1.78 104 90510 741 32 0.0431849 1.1.1.1.79 105 90930 742 32 0.0431267 1.1.1.1.80 106 92190 746 32 0.0428954 1.1.1.1.81 107 93030 747 32 0.042838 1.1.1.1.82 108 94290 750 32 0.0426667 1.1.1.1.83 109 95970 752 32 0.0425532 1.1.1.1.84 110 96810 755 32 0.0423841 1.1.1.1.85 111 97230 756 32 0.042328 1.1.1.1.86 112 98070 757 32 0.0422721 1.1.1.1.87 113 100590 763 32 0.0419397 1.1.1.1.88 114 102270 767 32 0.041721 1.1.1.1.89 115 103110 769 32 0.0416125 1.1.1.1.90 116 104790 770 32 0.0415584 1.1.1.1.91 117 105630 772 32 0.0414508 1.1.1.1.92 118 106890 773 32 0.0413972 1.1.1.1.93 119 108570 1547 64 0.0413704 1.1.1.1.1.10 120 109410 780 32 0.0410256 1.1.1.1.94 121 109830 781 32 0.0409731 1.1.1.1.95 122 113610 788 32 0.0406091 1.1.1.1.96 123 114870 789 32 0.0405577 1.1.1.1.97 124 116970 792 32 0.040404 1.1.1.1.98 125 118230 796 32 0.040201 1.1.1.1.99 126 119490 797 32 0.0401506 1.1.1.1.100 127 121170 800 32 0.04 1.1.1.1.102 128 123270 804 32 0.039801 1.1.1.1.103 129 124530 806 32 0.0397022 1.1.1.1.104 130 125790 808 32 0.039604 1.1.1.1.105 131 126210 809 32 0.039555 1.1.1.1.106 132 127470 810 32 0.0395062 1.1.1.1.107 133 128730 813 32 0.0393604 1.1.1.1.108 134 129570 814 32 0.039312 1.1.1.1.109 135 129990 816 32 0.0392157 1.1.1.1.110 136 132510 821 32 0.0389769 1.1.1.1.111 137 134610 823 32 0.0388821 1.1.1.1.112 138 135030 824 32 0.038835 1.1.1.1.113 139 137130 825 32 0.0387879 1.1.1.1.115 140 138390 828 32 0.0386473 1.1.1.1.116 141 141330 833 32 0.0384154 1.1.1.1.118 142 142170 834 32 0.0383693 1.1.1.1.119 143 143430 836 32 0.0382775 1.1.1.1.120 144 145110 838 32 0.0381862 1.1.1.1.121 145 147210 841 32 0.0380499 1.1.1.1.122 146 148890 843 32 0.0379597 1.1.1.1.123 147 150990 845 32 0.0378698 1.1.1.1.124 148 152670 848 32 0.0377358 1.1.1.1.125 149 153930 852 32 0.0375587 1.1.1.1.126 150 155190 853 32 0.0375147 1.1.1.1.127 151 156030 854 32 0.0374707 1.1.1.1.128 152 157710 858 32 0.037296 1.1.1.1.129 153 158970 859 32 0.0372526 1.1.1.1.130 154 161490 861 32 0.0371661 1.1.1.1.132 155 162330 862 32 0.037123 1.1.1.1.133 156 165270 866 32 0.0369515 1.1.1.1.134 157 167370 869 32 0.0368239 1.1.1.1.135 158 169890 873 32 0.0366552 1.1.1.1.136 159 170310 874 32 0.0366133 1.1.1.1.137 160 172410 876 32 0.0365297 1.1.1.1.138 161 172830 877 32 0.036488 1.1.1.1.139 162 173670 878 32 0.0364465 1.1.1.1.140 163 176190 881 32 0.0363224 1.1.1.1.142 164 179130 884 32 0.0361991 1.1.1.1.143 165 179970 885 32 0.0361582 1.1.1.1.144 166 180390 887 32 0.0360767 1.1.1.1.145 167 184170 891 32 0.0359147 1.1.1.1.147 168 185010 892 32 0.0358744 1.1.1.1.148 169 185430 893 32 0.0358343 1.1.1.1.149 170 190470 897 32 0.0356745 1.1.1.1.151 171 191310 898 32 0.0356347 1.1.1.1.152 172 192990 900 32 0.0355556 1.1.1.1.153 173 195090 903 32 0.0354374 1.1.1.1.154 174 196770 905 32 0.0353591 1.1.1.1.155 175 197610 908 32 0.0352423 1.1.1.1.156 176 198870 909 32 0.0352035 1.1.1.1.157 177 200130 910 32 0.0351648 1.1.1.1.158 178 203070 914 32 0.0350109 1.1.1.1.159 179 205170 916 32 0.0349345 1.1.1.1.161 180 206430 917 32 0.0348964 1.1.1.1.162 181 208110 918 32 0.0348584 1.1.1.1.163 182 211890 921 32 0.0347448 1.1.1.1.165 183 212730 922 32 0.0347072 1.1.1.1.166 184 214410 924 32 0.034632 1.1.1.1.168 185 216510 927 32 0.03452 1.1.1.1.169 186 218190 928 32 0.0344828 1.1.1.1.171 187 220290 931 32 0.0343716 1.1.1.1.172 188 220710 932 32 0.0343348 1.1.1.1.173 189 222810 934 32 0.0342612 1.1.1.1.174 190 224490 935 32 0.0342246 1.1.1.1.176 191 228270 938 32 0.0341151 1.1.1.1.177 192 229530 939 32 0.0340788 1.1.1.1.179 193 230370 940 32 0.0340426 1.1.1.1.180 194 231630 943 32 0.0339343 1.1.1.1.181 195 234570 945 32 0.0338624 1.1.1.1.183 196 235830 947 32 0.0337909 1.1.1.1.184 197 237090 949 32 0.0337197 1.1.1.1.185 198 241710 952 32 0.0336134 1.1.1.1.186 199 242130 953 32 0.0335782 1.1.1.1.187 200 244230 954 32 0.033543 1.1.1.1.188 201 245910 957 32 0.0334378 1.1.1.1.189 202 248010 958 32 0.0334029 1.1.1.1.190 203 249270 959 32 0.0333681 1.1.1.1.191 204 250530 961 32 0.0332986 1.1.1.1.192 205 252210 964 32 0.033195 1.1.1.1.193 206 254730 966 32 0.0331263 1.1.1.1.194 207 255570 967 32 0.033092 1.1.1.1.195 208 256830 968 32 0.0330579 1.1.1.1.196 209 258090 970 32 0.0329897 1.1.1.1.197 210 259770 972 32 0.0329218 1.1.1.1.199 211 262290 973 32 0.032888 1.1.1.1.200 212 264390 976 32 0.0327869 1.1.1.1.201 213 268170 977 32 0.0327533 1.1.1.1.202 214 269430 979 32 0.0326864 1.1.1.1.204 215 270690 980 32 0.0326531 1.1.1.1.205 216 272370 981 32 0.0326198 1.1.1.1.207 217 273630 982 32 0.0325866 1.1.1.1.209 218 274470 983 32 0.0325534 1.1.1.1.210 219 276990 986 32 0.0324544 1.1.1.1.211 220 278670 987 32 0.0324215 1.1.1.1.213 221 285810 993 32 0.0322256 1.1.1.1.214 222 287070 994 32 0.0321932 1.1.1.1.215 223 288330 996 32 0.0321285 1.1.1.1.216 224 293790 998 32 0.0320641 1.1.1.1.218 225 295890 1001 32 0.031968 1.1.1.1.219 226 298830 1003 32 0.0319043 1.1.1.1.220 227 300090 1004 32 0.0318725 1.1.1.1.222 228 300930 1005 32 0.0318408 1.1.1.1.223 229 302190 1006 32 0.0318091 1.1.1.1.224 230 303870 1009 32 0.0317146 1.1.1.1.225 231 306390 1011 32 0.0316518 1.1.1.1.228 232 308910 1014 32 0.0315582 1.1.1.1.229 233 311430 1015 32 0.0315271 1.1.1.1.231 234 312690 1016 32 0.0314961 1.1.1.1.233 235 314790 1017 32 0.0314651 1.1.1.1.235 236 317310 1019 32 0.0314033 1.1.1.1.236 237 319830 1020 32 0.0313725 1.1.1.1.237 238 321510 1022 32 0.0313112 1.1.1.1.238 239 324030 1024 32 0.03125 1.1.1.1.239 240 325290 1025 32 0.0312195 1.1.1.1.240 241 327390 1026 32 0.0311891 1.1.1.1.242 242 329070 1029 32 0.0310982 1.1.1.1.243 243 329910 1030 32 0.031068 1.1.1.1.244 244 331590 1031 32 0.0310378 1.1.1.1.245 245 332430 1032 32 0.0310078 1.1.1.1.246 246 335370 1033 32 0.0309777 1.1.1.1.247 247 336210 1035 32 0.0309179 1.1.1.1.248 248 337890 1036 32 0.030888 1.1.1.1.250 249 338730 1037 32 0.0308582 1.1.1.1.251 250 340410 1038 32 0.0308285 1.1.1.1.253 251 343770 1039 32 0.0307988 1.1.1.1.255 252 347970 1042 32 0.0307102 1.1.1.1.256 253 350070 1044 32 0.0306513 1.1.1.1.258 254 355530 1049 32 0.0305052 1.1.1.1.260 255 358890 1051 32 0.0304472 1.1.1.1.263 256 361410 1053 32 0.0303894 1.1.1.1.264 257 363930 1054 32 0.0303605 1.1.1.1.266 258 365610 1055 32 0.0303318 1.1.1.1.267 259 368130 1057 32 0.0302744 1.1.1.1.269 260 369390 1058 32 0.0302457 1.1.1.1.270 261 373170 1059 32 0.0302172 1.1.1.1.271 262 374430 1060 32 0.0301887 1.1.1.1.272 263 375270 1061 32 0.0301602 1.1.1.1.273 264 378210 1063 32 0.0301035 1.1.1.1.275 265 382830 1065 32 0.0300469 1.1.1.1.277 266 384510 1067 32 0.0299906 1.1.1.1.278 267 387870 1069 32 0.0299345 1.1.1.1.279 268 390810 1072 32 0.0298507 1.1.1.1.280 269 392070 1073 32 0.0298229 1.1.1.1.281 270 393330 1074 32 0.0297952 1.1.1.1.283 271 394170 1076 32 0.0297398 1.1.1.1.284 272 396690 1077 32 0.0297122 1.1.1.1.286 273 399210 1078 32 0.0296846 1.1.1.1.287 274 400470 1079 32 0.0296571 1.1.1.1.288 275 401730 1080 32 0.0296296 1.1.1.1.289 276 405510 1083 32 0.0295476 1.1.1.1.290 277 409290 1084 32 0.0295203 1.1.1.1.292 278 409710 1085 32 0.0294931 1.1.1.1.293 279 414330 1087 32 0.0294388 1.1.1.1.294 280 415590 1088 32 0.0294118 1.1.1.1.295 281 417270 1089 32 0.0293848 1.1.1.1.296 282 420630 1092 32 0.029304 1.1.1.1.300 283 422310 1093 32 0.0292772 1.1.1.1.301 284 423570 1094 32 0.0292505 1.1.1.1.302 285 425670 1095 32 0.0292237 1.1.1.1.303 286 431130 1098 32 0.0291439 1.1.1.1.306 287 433230 1100 32 0.0290909 1.1.1.1.307 288 437010 1101 32 0.0290645 1.1.1.1.309 289 437430 1102 32 0.0290381 1.1.1.1.310 290 438270 1103 32 0.0290118 1.1.1.1.311 291 440790 1104 32 0.0289855 1.1.1.1.313 292 443310 1106 32 0.0289331 1.1.1.1.314 293 447090 1107 32 0.028907 1.1.1.1.316 294 448770 1108 32 0.0288809 1.1.1.1.318 295 450030 1109 32 0.0288548 1.1.1.1.320 296 452130 1110 32 0.0288288 1.1.1.1.321 297 453810 1112 32 0.028777 1.1.1.1.322 298 457590 1113 32 0.0287511 1.1.1.1.323 299 459690 2229 64 0.0287124 1.1.1.1.1.41 300 462630 1118 32 0.0286225 1.1.1.1.324 301 463470 1119 32 0.028597 1.1.1.1.325 302 469770 1121 32 0.0285459 1.1.1.1.328 303 471030 1123 32 0.0284951 1.1.1.1.330 304 472710 1125 32 0.0284444 1.1.1.1.331 305 476490 1126 32 0.0284192 1.1.1.1.333 306 480270 1128 32 0.0283688 1.1.1.1.336 307 484890 1129 32 0.0283437 1.1.1.1.339 308 487410 2262 64 0.0282935 1.1.1.1.1.42 309 491190 1132 32 0.0282686 1.1.1.1.342 310 491610 1133 32 0.0282436 1.1.1.1.343 311 492870 1134 32 0.0282187 1.1.1.1.344 312 494970 1135 32 0.0281938 1.1.1.1.346 313 497910 1137 32 0.0281442 1.1.1.1.347 314 500010 1138 32 0.0281195 1.1.1.1.349 315 500430 1139 32 0.0280948 1.1.1.1.350 316 502530 1140 32 0.0280702 1.1.1.1.352 317 506310 1143 32 0.0279965 1.1.1.1.354 318 508830 1144 32 0.027972 1.1.1.1.356 319 510510 72 4843 128 0.0264299 1.1.1.1.1.1.1 320 570570 73 4939 128 0.0259162 1.1.1.1.1.1.2 321 690690 74 5119 128 0.0250049 1.1.1.1.1.1.3 322 746130 75 5138 128 0.0249124 1.1.1.1.1.2.1 323 801570 2570 64 0.0249027 1.1.1.1.1.64 324 806190 2573 64 0.0248737 1.1.1.1.1.65 325 815430 2580 64 0.0248062 1.1.1.1.1.66 326 829290 2592 64 0.0246914 1.1.1.1.1.67 327 847770 2608 64 0.0245399 1.1.1.1.1.68 328 861630 2617 64 0.0244555 1.1.1.1.1.69 329 870870 76 5364 128 0.0238628 1.1.1.1.1.1.4 330 930930 77 5436 128 0.0235467 1.1.1.1.1.1.5 331 1000230 2721 64 0.0235208 1.1.1.1.1.79 332 1014090 2730 64 0.0234432 1.1.1.1.1.80 333 1023330 2735 64 0.0234004 1.1.1.1.1.81 334 1037190 2744 64 0.0233236 1.1.1.1.1.82 335 1055670 2755 64 0.0232305 1.1.1.1.1.83 336 1064910 2765 64 0.0231465 1.1.1.1.1.84 337 1069530 2766 64 0.0231381 1.1.1.1.1.85 338 1078770 2772 64 0.023088 1.1.1.1.1.86 339 1106490 2791 64 0.0229308 1.1.1.1.1.87 340 1111110 5645 128 0.0226749 1.1.1.1.1.1.6 341 1175790 2830 64 0.0226148 1.1.1.1.1.92 342 1203510 2851 64 0.0224483 1.1.1.1.1.93 343 1208130 2853 64 0.0224325 1.1.1.1.1.94 344 1231230 5775 128 0.0221645 1.1.1.1.1.1.7 345 1286670 2898 64 0.0220842 1.1.1.1.1.97 346 1291290 5832 128 0.0219479 1.1.1.1.1.1.8 347 1319010 2917 64 0.0219403 1.1.1.1.1.100 348 1332870 2925 64 0.0218803 1.1.1.1.1.101 349 1355970 2936 64 0.0217984 1.1.1.1.1.102 350 1369830 2944 64 0.0217391 1.1.1.1.1.103 351 1383690 2953 64 0.0216729 1.1.1.1.1.104 352 1388310 2954 64 0.0216655 1.1.1.1.1.105 353 1402170 2960 64 0.0216216 1.1.1.1.1.106 354 1411410 5951 128 0.021509 1.1.1.1.1.1.9 355 1429890 2976 64 0.0215054 1.1.1.1.1.109 356 1457610 2991 64 0.0213975 1.1.1.1.1.110 357 1480710 3001 64 0.0213262 1.1.1.1.1.111 358 1485330 3004 64 0.0213049 1.1.1.1.1.112 359 1494570 3009 64 0.0212695 1.1.1.1.1.113 360 1508430 3015 64 0.0212272 1.1.1.1.1.114 361 1522290 3024 64 0.021164 1.1.1.1.1.115 362 1526910 3026 64 0.02115 1.1.1.1.1.116 363 1554630 3040 64 0.0210526 1.1.1.1.1.117 364 1563870 3044 64 0.021025 1.1.1.1.1.118 365 1577730 3052 64 0.0209699 1.1.1.1.1.119 366 1591590 6111 128 0.0209458 1.1.1.1.1.1.10 367 1596210 3060 64 0.020915 1.1.1.1.1.120 368 1619310 3072 64 0.0208333 1.1.1.1.1.121 369 1637790 3080 64 0.0207792 1.1.1.1.1.122 370 1660890 3093 64 0.0206919 1.1.1.1.1.123 371 1679370 3100 64 0.0206452 1.1.1.1.1.124 372 1693230 3108 64 0.020592 1.1.1.1.1.125 373 1707090 3113 64 0.0205589 1.1.1.1.1.126 374 1716330 3118 64 0.020526 1.1.1.1.1.127 375 1734810 3127 64 0.0204669 1.1.1.1.1.128 376 1748670 3131 64 0.0204408 1.1.1.1.1.129 377 1757910 3136 64 0.0204082 1.1.1.1.1.130 378 1776390 3146 64 0.0203433 1.1.1.1.1.131 379 1785630 3148 64 0.0203304 1.1.1.1.1.132 380 1817970 3162 64 0.0202404 1.1.1.1.1.133 381 1841070 3172 64 0.0201765 1.1.1.1.1.134 382 1868790 3185 64 0.0200942 1.1.1.1.1.135 383 1873410 3186 64 0.0200879 1.1.1.1.1.136 384 1896510 3196 64 0.020025 1.1.1.1.1.137 385 1901130 3198 64 0.0200125 1.1.1.1.1.138 386 1910370 3201 64 0.0199938 1.1.1.1.1.139 387 1914990 3202 64 0.0199875 1.1.1.1.1.140 388 1938090 3214 64 0.0199129 1.1.1.1.1.141 389 1970430 3229 64 0.0198204 1.1.1.1.1.142 390 1979670 3232 64 0.019802 1.1.1.1.1.143 391 1984290 3235 64 0.0197836 1.1.1.1.1.144 392 1993530 3238 64 0.0197653 1.1.1.1.1.145 393 2025870 3250 64 0.0196923 1.1.1.1.1.146 394 2035110 3254 64 0.0196681 1.1.1.1.1.147 395 2039730 3255 64 0.0196621 1.1.1.1.1.148 396 2048970 3259 64 0.0196379 1.1.1.1.1.149 397 2095170 3274 64 0.019548 1.1.1.1.1.150 398 2104410 3280 64 0.0195122 1.1.1.1.1.151 399 2122890 3289 64 0.0194588 1.1.1.1.1.152 400 2145990 3298 64 0.0194057 1.1.1.1.1.153 401 2164470 3306 64 0.0193587 1.1.1.1.1.154 402 2173710 3310 64 0.0193353 1.1.1.1.1.155 403 2187570 3315 64 0.0193062 1.1.1.1.1.156 404 2201430 3319 64 0.0192829 1.1.1.1.1.157 405 2233770 3330 64 0.0192192 1.1.1.1.1.158 406 2243010 3334 64 0.0191962 1.1.1.1.1.159 407 2256870 3339 64 0.0191674 1.1.1.1.1.160 408 2270730 3345 64 0.019133 1.1.1.1.1.161 409 2289210 3351 64 0.0190988 1.1.1.1.1.162 410 2303070 3355 64 0.019076 1.1.1.1.1.163 411 2330790 3365 64 0.0190193 1.1.1.1.1.164 412 2340030 3368 64 0.0190024 1.1.1.1.1.165 413 2353890 3374 64 0.0189686 1.1.1.1.1.166 414 2358510 3376 64 0.0189573 1.1.1.1.1.167 415 2381610 3386 64 0.0189014 1.1.1.1.1.168 416 2386230 3388 64 0.0188902 1.1.1.1.1.169 417 2400090 3391 64 0.0188735 1.1.1.1.1.170 418 2423190 3401 64 0.018818 1.1.1.1.1.171 419 2427810 3402 64 0.0188125 1.1.1.1.1.172 420 2450910 3409 64 0.0187738 1.1.1.1.1.173 421 2469390 3414 64 0.0187463 1.1.1.1.1.175 422 2510970 3430 64 0.0186589 1.1.1.1.1.176 423 2520210 3433 64 0.0186426 1.1.1.1.1.177 424 2524830 3435 64 0.0186317 1.1.1.1.1.178 425 2534070 3437 64 0.0186209 1.1.1.1.1.179 426 2547930 3442 64 0.0185938 1.1.1.1.1.180 427 2561790 3446 64 0.0185723 1.1.1.1.1.181 428 2580270 3452 64 0.01854 1.1.1.1.1.182 429 2594130 3458 64 0.0185078 1.1.1.1.1.183 430 2607990 3461 64 0.0184918 1.1.1.1.1.184 431 2658810 3479 64 0.0183961 1.1.1.1.1.185 432 2663430 3481 64 0.0183855 1.1.1.1.1.186 433 2686530 3485 64 0.0183644 1.1.1.1.1.187 434 2705010 3493 64 0.0183224 1.1.1.1.1.188 435 2728110 3500 64 0.0182857 1.1.1.1.1.189 436 2741970 3504 64 0.0182648 1.1.1.1.1.190 437 2755830 3510 64 0.0182336 1.1.1.1.1.191 438 2774310 3517 64 0.0181973 1.1.1.1.1.192 439 2802030 3524 64 0.0181612 1.1.1.1.1.193 440 2811270 3527 64 0.0181457 1.1.1.1.1.194 441 2825130 3532 64 0.01812 1.1.1.1.1.195 442 2838990 3536 64 0.0180995 1.1.1.1.1.196 443 2857470 3539 64 0.0180842 1.1.1.1.1.198 444 2885190 3548 64 0.0180383 1.1.1.1.1.199 445 2908290 3556 64 0.0179978 1.1.1.1.1.200 446 2949870 3568 64 0.0179372 1.1.1.1.1.201 447 2954490 3570 64 0.0179272 1.1.1.1.1.202 448 2963730 3573 64 0.0179121 1.1.1.1.1.203 449 2977590 3577 64 0.0178921 1.1.1.1.1.204 450 2982210 3578 64 0.0178871 1.1.1.1.1.205 451 2996070 3582 64 0.0178671 1.1.1.1.1.206 452 3005310 3585 64 0.0178522 1.1.1.1.1.207 453 3009930 3586 64 0.0178472 1.1.1.1.1.208 454 3019170 3589 64 0.0178323 1.1.1.1.1.209 455 3046890 3597 64 0.0177926 1.1.1.1.1.210 456 3051510 3599 64 0.0177827 1.1.1.1.1.211 457 3065370 3602 64 0.0177679 1.1.1.1.1.212 458 3143910 3623 64 0.0176649 1.1.1.1.1.213 459 3157770 3627 64 0.0176454 1.1.1.1.1.214 460 3171630 3631 64 0.017626 1.1.1.1.1.215 461 3190110 3637 64 0.0175969 1.1.1.1.1.216 462 3231690 3649 64 0.0175391 1.1.1.1.1.217 463 3254790 3656 64 0.0175055 1.1.1.1.1.218 464 3287130 3665 64 0.0174625 1.1.1.1.1.219 465 3296370 3669 64 0.0174434 1.1.1.1.1.220 466 3300990 3670 64 0.0174387 1.1.1.1.1.221 467 3310230 3673 64 0.0174244 1.1.1.1.1.222 468 3324090 3676 64 0.0174102 1.1.1.1.1.223 469 3342570 3681 64 0.0173866 1.1.1.1.1.224 470 3351810 3682 64 0.0173819 1.1.1.1.1.225 471 3356430 3683 64 0.0173771 1.1.1.1.1.226 472 3370290 3689 64 0.0173489 1.1.1.1.1.227 473 3398010 3696 64 0.017316 1.1.1.1.1.228 474 3421110 3700 64 0.0172973 1.1.1.1.1.229 475 3425730 3701 64 0.0172926 1.1.1.1.1.230 476 3434970 3704 64 0.0172786 1.1.1.1.1.231 477 3439590 3705 64 0.017274 1.1.1.1.1.232 478 3448830 3707 64 0.0172646 1.1.1.1.1.233 479 3462690 3711 64 0.017246 1.1.1.1.1.234 480 3490410 3718 64 0.0172136 1.1.1.1.1.235 481 3518130 3726 64 0.0171766 1.1.1.1.1.236 482 3536610 3732 64 0.017149 1.1.1.1.1.237 483 3564330 3741 64 0.0171077 1.1.1.1.1.238 484 3578190 3744 64 0.017094 1.1.1.1.1.239 485 3587430 3746 64 0.0170849 1.1.1.1.1.240 486 3601290 3749 64 0.0170712 1.1.1.1.1.241 487 3619770 3754 64 0.0170485 1.1.1.1.1.242 488 3629010 3756 64 0.0170394 1.1.1.1.1.243 489 3647490 3760 64 0.0170213 1.1.1.1.1.244 490 3656730 3762 64 0.0170122 1.1.1.1.1.245 491 3689070 3771 64 0.0169716 1.1.1.1.1.246 492 3698310 3774 64 0.0169581 1.1.1.1.1.247 493 3712170 3777 64 0.0169447 1.1.1.1.1.248 494 3716790 3778 64 0.0169402 1.1.1.1.1.249 495 3726030 3781 64 0.0169267 1.1.1.1.1.250 496 3739890 3784 64 0.0169133 1.1.1.1.1.251 497 3744510 3785 64 0.0169089 1.1.1.1.1.252 498 3758370 3788 64 0.0168955 1.1.1.1.1.253 499 3781470 3796 64 0.0168599 1.1.1.1.1.254 500 3827670 3805 64 0.01682 1.1.1.1.1.255 501 3841530 3808 64 0.0168067 1.1.1.1.1.256 502 3850770 3812 64 0.0167891 1.1.1.1.1.257 503 3910830 3827 64 0.0167233 1.1.1.1.1.259 504 3920070 3830 64 0.0167102 1.1.1.1.1.260 505 3924690 3831 64 0.0167058 1.1.1.1.1.261 506 3947790 3836 64 0.016684 1.1.1.1.1.262 507 3975510 3843 64 0.0166537 1.1.1.1.1.263 508 4003230 3850 64 0.0166234 1.1.1.1.1.265 509 4021710 3852 64 0.0166147 1.1.1.1.1.266 510 4035570 3856 64 0.0165975 1.1.1.1.1.267 511 4049430 3858 64 0.0165889 1.1.1.1.1.268 512 4063290 3860 64 0.0165803 1.1.1.1.1.269 513 4104870 3871 64 0.0165332 1.1.1.1.1.270 514 4118730 3876 64 0.0165119 1.1.1.1.1.271 515 4127970 3877 64 0.0165076 1.1.1.1.1.272 516 4132590 3878 64 0.0165034 1.1.1.1.1.273 517 4160310 3886 64 0.0164694 1.1.1.1.1.274 518 4183410 3889 64 0.0164567 1.1.1.1.1.275 519 4211130 3896 64 0.0164271 1.1.1.1.1.276 520 4229610 3901 64 0.016406 1.1.1.1.1.277 521 4266570 3910 64 0.0163683 1.1.1.1.1.278 522 4298910 3916 64 0.0163432 1.1.1.1.1.279 523 4312770 3921 64 0.0163224 1.1.1.1.1.280 524 4322010 3923 64 0.016314 1.1.1.1.1.281 525 4326630 3925 64 0.0163057 1.1.1.1.1.282 526 4335870 3927 64 0.0162974 1.1.1.1.1.283 527 4340490 3928 64 0.0162933 1.1.1.1.1.284 528 4363590 3933 64 0.0162726 1.1.1.1.1.285 529 4391310 3938 64 0.0162519 1.1.1.1.1.286 530 4405170 3940 64 0.0162437 1.1.1.1.1.287 531 4419030 3944 64 0.0162272 1.1.1.1.1.288 532 4460610 3952 64 0.0161943 1.1.1.1.1.289 533 4465230 3953 64 0.0161902 1.1.1.1.1.290 534 4502190 3962 64 0.0161535 1.1.1.1.1.291 535 4506810 3963 64 0.0161494 1.1.1.1.1.292 536 4557630 3973 64 0.0161087 1.1.1.1.1.293 537 4571490 3975 64 0.0161006 1.1.1.1.1.294 538 4589970 3979 64 0.0160844 1.1.1.1.1.295 539 4603830 3982 64 0.0160723 1.1.1.1.1.296 540 4613070 3985 64 0.0160602 1.1.1.1.1.297 541 4626930 3988 64 0.0160481 1.1.1.1.1.299 542 4645410 3991 64 0.0160361 1.1.1.1.1.300 543 4659270 3995 64 0.01602 1.1.1.1.1.301 544 4682370 3998 64 0.016008 1.1.1.1.1.302 545 4686990 3999 64 0.016004 1.1.1.1.1.303 546 4710090 4004 64 0.015984 1.1.1.1.1.304 547 4742430 4011 64 0.0159561 1.1.1.1.1.305 548 4765530 4018 64 0.0159283 1.1.1.1.1.306 549 4779390 4019 64 0.0159244 1.1.1.1.1.307 550 4807110 4025 64 0.0159006 1.1.1.1.1.308 551 4811730 4026 64 0.0158967 1.1.1.1.1.309 552 4820970 4028 64 0.0158888 1.1.1.1.1.310 553 4825590 4029 64 0.0158848 1.1.1.1.1.311 554 4848690 4034 64 0.0158651 1.1.1.1.1.312 555 4876410 4039 64 0.0158455 1.1.1.1.1.313 556 4881030 4040 64 0.0158416 1.1.1.1.1.314 557 4917990 4047 64 0.0158142 1.1.1.1.1.315 558 4922610 4050 64 0.0158025 1.1.1.1.1.316 559 4936470 4051 64 0.0157986 1.1.1.1.1.317 560 4945710 4054 64 0.0157869 1.1.1.1.1.318 561 4950330 4055 64 0.015783 1.1.1.1.1.319 562 4973430 4059 64 0.0157674 1.1.1.1.1.320 563 4991910 4064 64 0.015748 1.1.1.1.1.321 564 5033490 4071 64 0.015721 1.1.1.1.1.322 565 5088930 4082 64 0.0156786 1.1.1.1.1.323 566 5098170 4083 64 0.0156747 1.1.1.1.1.324 567 5112030 4085 64 0.0156671 1.1.1.1.1.325 568 5130510 4088 64 0.0156556 1.1.1.1.1.326 569 5167470 4095 64 0.0156288 1.1.1.1.1.327 570 5181330 4098 64 0.0156174 1.1.1.1.1.329 571 5199810 4103 64 0.0155983 1.1.1.1.1.330 572 5236770 4108 64 0.0155794 1.1.1.1.1.331 573 5241390 4109 64 0.0155756 1.1.1.1.1.332 574 5250630 4112 64 0.0155642 1.1.1.1.1.333 575 5269110 4114 64 0.0155566 1.1.1.1.1.334 576 5282970 4118 64 0.0155415 1.1.1.1.1.335 577 5296830 4122 64 0.0155264 1.1.1.1.1.336 578 5333790 4129 64 0.0155001 1.1.1.1.1.338 579 5338410 4130 64 0.0154964 1.1.1.1.1.339 580 5389230 4137 64 0.0154701 1.1.1.1.1.340 581 5403090 4141 64 0.0154552 1.1.1.1.1.341 582 5407710 4142 64 0.0154515 1.1.1.1.1.342 583 5421570 4145 64 0.0154403 1.1.1.1.1.343 584 5444670 4147 64 0.0154328 1.1.1.1.1.345 585 5477010 4154 64 0.0154068 1.1.1.1.1.346 586 5490870 4158 64 0.015392 1.1.1.1.1.347 587 5500110 4159 64 0.0153883 1.1.1.1.1.348 588 5504730 4160 64 0.0153846 1.1.1.1.1.349 589 5518590 4163 64 0.0153735 1.1.1.1.1.350 590 5527830 4164 64 0.0153698 1.1.1.1.1.351 591 5541690 4167 64 0.0153588 1.1.1.1.1.352 592 5569410 4172 64 0.0153404 1.1.1.1.1.353 593 5583270 4173 64 0.0153367 1.1.1.1.1.354 594 5597130 4175 64 0.0153293 1.1.1.1.1.355 595 5629470 4182 64 0.0153037 1.1.1.1.1.356 596 5638710 4184 64 0.0152964 1.1.1.1.1.357 597 5652570 4187 64 0.0152854 1.1.1.1.1.358 598 5680290 4192 64 0.0152672 1.1.1.1.1.359 599 5698770 4193 64 0.0152635 1.1.1.1.1.360 600 5712630 4195 64 0.0152563 1.1.1.1.1.361 601 5721870 4197 64 0.015249 1.1.1.1.1.362 602 5781930 4208 64 0.0152091 1.1.1.1.1.363 603 5823510 4218 64 0.0151731 1.1.1.1.1.364 604 5846610 4221 64 0.0151623 1.1.1.1.1.365 605 5865090 4224 64 0.0151515 1.1.1.1.1.366 606 5874330 4225 64 0.0151479 1.1.1.1.1.367 607 5888190 4229 64 0.0151336 1.1.1.1.1.368 608 5906670 4233 64 0.0151193 1.1.1.1.1.370 609 5957490 4242 64 0.0150872 1.1.1.1.1.371 610 5985210 4245 64 0.0150766 1.1.1.1.1.372 611 5989830 4247 64 0.0150695 1.1.1.1.1.373 612 6026790 4253 64 0.0150482 1.1.1.1.1.374 613 6045270 4255 64 0.0150411 1.1.1.1.1.375 614 6054510 4258 64 0.0150305 1.1.1.1.1.376 615 6082230 4262 64 0.0150164 1.1.1.1.1.377 616 6114570 4267 64 0.0149988 1.1.1.1.1.378 617 6137670 4271 64 0.0149848 1.1.1.1.1.379 618 6151530 4274 64 0.0149743 1.1.1.1.1.381 619 6170010 4276 64 0.0149673 1.1.1.1.1.382 620 6183870 4278 64 0.0149603 1.1.1.1.1.383 621 6197730 4281 64 0.0149498 1.1.1.1.1.384 622 6206970 4282 64 0.0149463 1.1.1.1.1.385 623 6211590 4283 64 0.0149428 1.1.1.1.1.386 624 6220830 4284 64 0.0149393 1.1.1.1.1.387 625 6234690 4286 64 0.0149323 1.1.1.1.1.388 626 6253170 4289 64 0.0149219 1.1.1.1.1.389 627 6262410 4290 64 0.0149184 1.1.1.1.1.390 628 6280890 4292 64 0.0149115 1.1.1.1.1.392 629 6303990 4298 64 0.0148906 1.1.1.1.1.393 630 6308610 4299 64 0.0148872 1.1.1.1.1.394 631 6331710 4302 64 0.0148768 1.1.1.1.1.395 632 6350190 4304 64 0.0148699 1.1.1.1.1.396 633 6359430 4308 64 0.0148561 1.1.1.1.1.397 634 6391770 4313 64 0.0148389 1.1.1.1.1.398 635 6414870 4315 64 0.014832 1.1.1.1.1.399 636 6442590 4321 64 0.0148114 1.1.1.1.1.400 637 6461070 4324 64 0.0148011 1.1.1.1.1.402 638 6470310 4326 64 0.0147943 1.1.1.1.1.403 639 6474930 4327 64 0.0147908 1.1.1.1.1.404 640 6511890 4333 64 0.0147704 1.1.1.1.1.405 641 6544230 4337 64 0.0147567 1.1.1.1.1.406 642 6553470 4338 64 0.0147533 1.1.1.1.1.407 643 6567330 4342 64 0.0147398 1.1.1.1.1.408 644 6585810 4343 64 0.0147364 1.1.1.1.1.409 645 6599670 4346 64 0.0147262 1.1.1.1.1.410 646 6608910 4348 64 0.0147194 1.1.1.1.1.411 647 6650490 4354 64 0.0146991 1.1.1.1.1.412 648 6668970 4358 64 0.0146856 1.1.1.1.1.413 649 6692070 4361 64 0.0146755 1.1.1.1.1.414 650 6719790 4363 64 0.0146688 1.1.1.1.1.416 651 6738270 4368 64 0.014652 1.1.1.1.1.417 652 6761370 4372 64 0.0146386 1.1.1.1.1.418 653 6789090 4375 64 0.0146286 1.1.1.1.1.419 654 6821430 4381 64 0.0146085 1.1.1.1.1.420 655 6830670 4382 64 0.0146052 1.1.1.1.1.421 656 6844530 4384 64 0.0145985 1.1.1.1.1.422 657 6858390 4385 64 0.0145952 1.1.1.1.1.423 658 6863010 4388 64 0.0145852 1.1.1.1.1.424 659 6927690 4396 64 0.0145587 1.1.1.1.1.425 660 6932310 4398 64 0.0145521 1.1.1.1.1.426 661 6955410 4400 64 0.0145455 1.1.1.1.1.427 662 6973890 4403 64 0.0145355 1.1.1.1.1.428 663 7015470 4408 64 0.0145191 1.1.1.1.1.430 664 7024710 4410 64 0.0145125 1.1.1.1.1.431 665 7043190 4413 64 0.0145026 1.1.1.1.1.432 666 7070910 4417 64 0.0144895 1.1.1.1.1.433 667 7084770 4420 64 0.0144796 1.1.1.1.1.434 668 7112490 4425 64 0.0144633 1.1.1.1.1.435 669 7121730 4427 64 0.0144567 1.1.1.1.1.436 670 7135590 4429 64 0.0144502 1.1.1.1.1.437 671 7181790 4434 64 0.0144339 1.1.1.1.1.438 672 7204890 4439 64 0.0144177 1.1.1.1.1.439 673 7209510 4440 64 0.0144144 1.1.1.1.1.440 674 7246470 4444 64 0.0144014 1.1.1.1.1.441 675 7306530 4453 64 0.0143723 1.1.1.1.1.442 676 7320390 4454 64 0.0143691 1.1.1.1.1.444 677 7348110 4457 64 0.0143594 1.1.1.1.1.445 678 7361970 4460 64 0.0143498 1.1.1.1.1.446 679 7398930 4466 64 0.0143305 1.1.1.1.1.448 680 7412790 4470 64 0.0143177 1.1.1.1.1.449 681 7431270 4471 64 0.0143145 1.1.1.1.1.450 682 7440510 4473 64 0.0143081 1.1.1.1.1.451 683 7458990 4476 64 0.0142985 1.1.1.1.1.452 684 7509810 4484 64 0.014273 1.1.1.1.1.453 685 7528290 4486 64 0.0142666 1.1.1.1.1.456 686 7556010 4489 64 0.0142571 1.1.1.1.1.457 687 7620690 4500 64 0.0142222 1.1.1.1.1.458 688 7625310 4501 64 0.0142191 1.1.1.1.1.459 689 7653030 4502 64 0.0142159 1.1.1.1.1.461 690 7666890 4505 64 0.0142064 1.1.1.1.1.462 691 7689990 4509 64 0.0141938 1.1.1.1.1.464 692 7722330 4513 64 0.0141813 1.1.1.1.1.466 693 7731570 4515 64 0.014175 1.1.1.1.1.467 694 7759290 4518 64 0.0141656 1.1.1.1.1.468 695 7763910 4520 64 0.0141593 1.1.1.1.1.469 696 7787010 4523 64 0.0141499 1.1.1.1.1.470 697 7828590 4528 64 0.0141343 1.1.1.1.1.472 698 7870170 4534 64 0.0141156 1.1.1.1.1.474 699 7884030 4536 64 0.0141093 1.1.1.1.1.475 700 7930230 4543 64 0.0140876 1.1.1.1.1.476 701 7967190 4547 64 0.0140752 1.1.1.1.1.477 702 7985670 4550 64 0.0140659 1.1.1.1.1.478 703 7994910 4551 64 0.0140628 1.1.1.1.1.479 704 8008770 4554 64 0.0140536 1.1.1.1.1.481 705 8064210 4560 64 0.0140351 1.1.1.1.1.483 706 8082690 4561 64 0.014032 1.1.1.1.1.484 707 8110410 4567 64 0.0140136 1.1.1.1.1.485 708 8147370 4570 64 0.0140044 1.1.1.1.1.487 709 8151990 4571 64 0.0140013 1.1.1.1.1.488 710 8161230 4572 64 0.0139983 1.1.1.1.1.489 711 8175090 4575 64 0.0139891 1.1.1.1.1.490 712 8179710 4576 64 0.013986 1.1.1.1.1.491 713 8193570 4577 64 0.013983 1.1.1.1.1.492 714 8216670 4581 64 0.0139707 1.1.1.1.1.493 715 8221290 4582 64 0.0139677 1.1.1.1.1.494 716 8249010 4585 64 0.0139586 1.1.1.1.1.495 717 8272110 4590 64 0.0139434 1.1.1.1.1.496 718 8299830 4593 64 0.0139342 1.1.1.1.1.498 719 8332170 4598 64 0.0139191 1.1.1.1.1.499 720 8346030 4599 64 0.0139161 1.1.1.1.1.500 721 8355270 4601 64 0.01391 1.1.1.1.1.501 722 8369130 4602 64 0.013907 1.1.1.1.1.502 723 8387610 4604 64 0.013901 1.1.1.1.1.503 724 8401470 4607 64 0.0138919 1.1.1.1.1.504 725 8415330 4608 64 0.0138889 1.1.1.1.1.505 726 8452290 4615 64 0.0138678 1.1.1.1.1.506 727 8480010 4619 64 0.0138558 1.1.1.1.1.507 728 8493870 4620 64 0.0138528 1.1.1.1.1.509 729 8526210 4624 64 0.0138408 1.1.1.1.1.510 730 8540070 4625 64 0.0138378 1.1.1.1.1.511 731 8567790 4627 64 0.0138319 1.1.1.1.1.513 732 8590890 4630 64 0.0138229 1.1.1.1.1.514 733 8609370 4634 64 0.013811 1.1.1.1.1.515 734 8637090 4636 64 0.013805 1.1.1.1.1.517 735 8687910 4644 64 0.0137812 1.1.1.1.1.518 736 8701770 4646 64 0.0137753 1.1.1.1.1.519 737 8729490 4649 64 0.0137664 1.1.1.1.1.521 738 8761830 4654 64 0.0137516 1.1.1.1.1.522 739 8784930 4657 64 0.0137428 1.1.1.1.1.524 740 8826510 4663 64 0.0137251 1.1.1.1.1.525 741 8831130 4664 64 0.0137221 1.1.1.1.1.526 742 8854230 4665 64 0.0137192 1.1.1.1.1.527 743 8886570 4670 64 0.0137045 1.1.1.1.1.528 744 8895810 4672 64 0.0136986 1.1.1.1.1.529 745 8923530 4674 64 0.0136928 1.1.1.1.1.531 746 8955870 4676 64 0.0136869 1.1.1.1.1.532 747 8965110 4679 64 0.0136781 1.1.1.1.1.533 748 8983590 4680 64 0.0136752 1.1.1.1.1.534 749 9025170 4687 64 0.0136548 1.1.1.1.1.535 750 9034410 4688 64 0.0136519 1.1.1.1.1.536 751 9062130 4691 64 0.0136431 1.1.1.1.1.539 752 9075990 4694 64 0.0136344 1.1.1.1.1.540 753 9108330 4697 64 0.0136257 1.1.1.1.1.542 754 9117570 4698 64 0.0136228 1.1.1.1.1.543 755 9163770 4703 64 0.0136083 1.1.1.1.1.544 756 9214590 4708 64 0.0135939 1.1.1.1.1.545 757 9242310 4713 64 0.0135795 1.1.1.1.1.546 758 9246930 4714 64 0.0135766 1.1.1.1.1.547 759 9270030 4716 64 0.0135708 1.1.1.1.1.549 760 9283890 4717 64 0.0135679 1.1.1.1.1.550 761 9302370 4720 64 0.0135593 1.1.1.1.1.552 762 9353190 4725 64 0.013545 1.1.1.1.1.553 763 9357810 4726 64 0.0135421 1.1.1.1.1.554 764 9371670 4727 64 0.0135392 1.1.1.1.1.555 765 9408630 4730 64 0.0135307 1.1.1.1.1.556 766 9422490 4732 64 0.0135249 1.1.1.1.1.557 767 9450210 4737 64 0.0135107 1.1.1.1.1.558 768 9454830 4738 64 0.0135078 1.1.1.1.1.559 769 9468690 4739 64 0.013505 1.1.1.1.1.560 770 9496410 4742 64 0.0134964 1.1.1.1.1.561 771 9533370 4748 64 0.0134794 1.1.1.1.1.562 772 9547230 4749 64 0.0134765 1.1.1.1.1.564 773 9593430 4754 64 0.0134623 1.1.1.1.1.566 774 9602670 4755 64 0.0134595 1.1.1.1.1.567 775 9607290 4757 64 0.0134539 1.1.1.1.1.568 776 9648870 4762 64 0.0134397 1.1.1.1.1.569 777 9699690 104 19985 256 0.0128096 1.1.1.1.1.1.1.1 778 11741730 105 20605 256 0.0124242 1.1.1.1.1.1.1.2 779 13123110 106 20929 256 0.0122318 1.1.1.1.1.1.2.1 780 14430570 5236 64 0.0122231 1.1.1.1.1.807 781 14453670 5239 64 0.0122161 1.1.1.1.1.808 782 14467530 5240 64 0.0122137 1.1.1.1.1.809 783 14499870 5242 64 0.0122091 1.1.1.1.1.812 784 14522970 5245 64 0.0122021 1.1.1.1.1.813 785 14550690 5246 64 0.0121998 1.1.1.1.1.814 786 14555310 5248 64 0.0121951 1.1.1.1.1.815 787 14592270 5250 64 0.0121905 1.1.1.1.1.817 788 14606130 5251 64 0.0121882 1.1.1.1.1.818 789 14619990 5252 64 0.0121858 1.1.1.1.1.819 790 14638470 5253 64 0.0121835 1.1.1.1.1.820 791 14652330 5254 64 0.0121812 1.1.1.1.1.821 792 14675430 5256 64 0.0121766 1.1.1.1.1.822 793 14689290 5257 64 0.0121742 1.1.1.1.1.823 794 14707770 5261 64 0.012165 1.1.1.1.1.825 795 14758590 5264 64 0.0121581 1.1.1.1.1.828 796 14777070 5266 64 0.0121534 1.1.1.1.1.829 797 14804790 107 21453 256 0.0119331 1.1.1.1.1.1.1.3 798 15825810 108 21713 256 0.0117902 1.1.1.1.1.1.1.4 799 16546530 109 21769 256 0.0117598 1.1.1.1.1.1.2.2 800 17022390 5445 64 0.0117539 1.1.1.1.1.933 801 17077830 5448 64 0.0117474 1.1.1.1.1.934 802 17119410 5450 64 0.0117431 1.1.1.1.1.935 803 17170230 5453 64 0.0117367 1.1.1.1.1.937 804 17211810 5456 64 0.0117302 1.1.1.1.1.938 805 17225670 5458 64 0.0117259 1.1.1.1.1.939 806 17271870 5460 64 0.0117216 1.1.1.1.1.941 807 17281110 5461 64 0.0117195 1.1.1.1.1.942 808 17294970 5464 64 0.011713 1.1.1.1.1.943 809 17322690 5466 64 0.0117087 1.1.1.1.1.945 810 17341170 5467 64 0.0117066 1.1.1.1.1.946 811 17364270 5469 64 0.0117023 1.1.1.1.1.947 812 17378130 5470 64 0.0117002 1.1.1.1.1.948 813 17391990 5471 64 0.011698 1.1.1.1.1.949 814 17410470 5472 64 0.0116959 1.1.1.1.1.950 815 17419710 5473 64 0.0116938 1.1.1.1.1.951 816 17433570 5475 64 0.0116895 1.1.1.1.1.952 817 17465910 5477 64 0.0116852 1.1.1.1.1.955 818 17493630 5478 64 0.0116831 1.1.1.1.1.956 819 17502870 5480 64 0.0116788 1.1.1.1.1.957 820 17516730 5481 64 0.0116767 1.1.1.1.1.958 821 17562930 5483 64 0.0116724 1.1.1.1.1.961 822 17572170 5485 64 0.0116682 1.1.1.1.1.962 823 17604510 5487 64 0.0116639 1.1.1.1.1.963 824 17646090 5491 64 0.0116554 1.1.1.1.1.964 825 17655330 5492 64 0.0116533 1.1.1.1.1.965 826 17669190 5493 64 0.0116512 1.1.1.1.1.966 827 17687670 110 22028 256 0.0116216 1.1.1.1.1.1.2.3 828 17909430 5508 64 0.0116195 1.1.1.1.1.978 829 17918670 5509 64 0.0116174 1.1.1.1.1.979 830 17923290 5510 64 0.0116152 1.1.1.1.1.980 831 17992590 5514 64 0.0116068 1.1.1.1.1.981 832 18001830 5515 64 0.0116047 1.1.1.1.1.982 833 18057270 5521 64 0.0115921 1.1.1.1.1.983 834 18071130 5522 64 0.01159 1.1.1.1.1.984 835 18112710 5524 64 0.0115858 1.1.1.1.1.986 836 18140430 5525 64 0.0115837 1.1.1.1.1.987 837 18172770 5528 64 0.0115774 1.1.1.1.1.988 838 18195870 5530 64 0.0115732 1.1.1.1.1.990 839 18251310 5533 64 0.011567 1.1.1.1.1.993 840 18265170 5534 64 0.0115649 1.1.1.1.1.994 841 18292890 5535 64 0.0115628 1.1.1.1.1.995 842 18311370 5538 64 0.0115565 1.1.1.1.1.996 843 18362190 5540 64 0.0115523 1.1.1.1.1.999 844 18394530 5543 64 0.0115461 1.1.1.1.1.1001 845 18463830 5547 64 0.0115378 1.1.1.1.1.1002 846 18500790 5550 64 0.0115315 1.1.1.1.1.1003 847 18570090 5554 64 0.0115232 1.1.1.1.1.1006 848 18602430 5558 64 0.0115149 1.1.1.1.1.1007 849 18639390 5560 64 0.0115108 1.1.1.1.1.1009 850 18667110 5561 64 0.0115087 1.1.1.1.1.1010 851 18680970 5563 64 0.0115046 1.1.1.1.1.1011 852 18713310 5564 64 0.0115025 1.1.1.1.1.1014 853 18750270 5566 64 0.0114984 1.1.1.1.1.1016 854 18764130 5567 64 0.0114963 1.1.1.1.1.1017 855 18819570 5570 64 0.0114901 1.1.1.1.1.1018 856 18851910 5571 64 0.0114881 1.1.1.1.1.1019 857 18865770 5573 64 0.0114839 1.1.1.1.1.1020 858 18875010 5575 64 0.0114798 1.1.1.1.1.1021 859 18888870 111 22443 256 0.0114067 1.1.1.1.1.1.1.5 860 19373970 5612 64 0.0114041 1.1.1.1.1.1045 861 19447890 5616 64 0.011396 1.1.1.1.1.1047 862 19457130 5617 64 0.011394 1.1.1.1.1.1048 863 19470990 5618 64 0.011392 1.1.1.1.1.1049 864 19503330 5621 64 0.0113859 1.1.1.1.1.1051 865 19544910 5623 64 0.0113818 1.1.1.1.1.1053 866 19637310 5628 64 0.0113717 1.1.1.1.1.1055 867 19665030 5630 64 0.0113677 1.1.1.1.1.1056 868 19683510 5633 64 0.0113616 1.1.1.1.1.1057 869 19697370 5634 64 0.0113596 1.1.1.1.1.1058 870 19720470 5635 64 0.0113576 1.1.1.1.1.1059 871 19780530 5639 64 0.0113495 1.1.1.1.1.1062 872 19803630 5640 64 0.0113475 1.1.1.1.1.1063 873 19822110 5641 64 0.0113455 1.1.1.1.1.1064 874 19859070 5645 64 0.0113375 1.1.1.1.1.1065 875 19886790 5646 64 0.0113355 1.1.1.1.1.1067 876 19919130 5647 64 0.0113335 1.1.1.1.1.1068 877 19928370 5648 64 0.0113314 1.1.1.1.1.1069 878 19960710 5649 64 0.0113294 1.1.1.1.1.1071 879 19974570 5651 64 0.0113254 1.1.1.1.1.1072 880 20011530 5654 64 0.0113194 1.1.1.1.1.1073 881 20053110 5655 64 0.0113174 1.1.1.1.1.1076 882 20071590 5656 64 0.0113154 1.1.1.1.1.1077 883 20080830 5657 64 0.0113134 1.1.1.1.1.1078 884 20094690 5658 64 0.0113114 1.1.1.1.1.1079 885 20127030 5659 64 0.0113094 1.1.1.1.1.1081 886 20140890 5660 64 0.0113074 1.1.1.1.1.1082 887 20168610 5663 64 0.0113014 1.1.1.1.1.1083 888 20182470 5664 64 0.0112994 1.1.1.1.1.1084 889 20191710 5665 64 0.0112974 1.1.1.1.1.1085 890 20219430 5667 64 0.0112935 1.1.1.1.1.1087 891 20279490 5670 64 0.0112875 1.1.1.1.1.1089 892 20288730 5671 64 0.0112855 1.1.1.1.1.1090 893 20334930 5674 64 0.0112795 1.1.1.1.1.1091 894 20344170 5675 64 0.0112775 1.1.1.1.1.1092 895 20376510 5677 64 0.0112736 1.1.1.1.1.1094 896 20413470 5678 64 0.0112716 1.1.1.1.1.1096 897 20441190 5680 64 0.0112676 1.1.1.1.1.1098 898 20468910 5681 64 0.0112656 1.1.1.1.1.1099 899 20482770 5682 64 0.0112636 1.1.1.1.1.1101 900 20528970 5686 64 0.0112557 1.1.1.1.1.1102 901 20612130 5691 64 0.0112458 1.1.1.1.1.1104 902 20625990 5693 64 0.0112419 1.1.1.1.1.1105 903 20653710 5694 64 0.0112399 1.1.1.1.1.1107 904 20676810 5696 64 0.011236 1.1.1.1.1.1108 905 20704530 5698 64 0.011232 1.1.1.1.1.1109 906 20718390 5699 64 0.01123 1.1.1.1.1.1110 907 20787690 5703 64 0.0112222 1.1.1.1.1.1112 908 20792310 5704 64 0.0112202 1.1.1.1.1.1113 909 20806170 5705 64 0.0112182 1.1.1.1.1.1114 910 20820030 5706 64 0.0112163 1.1.1.1.1.1116 911 20856990 5707 64 0.0112143 1.1.1.1.1.1117 912 20884710 5708 64 0.0112123 1.1.1.1.1.1118 913 20903190 5709 64 0.0112104 1.1.1.1.1.1120 914 20926290 5711 64 0.0112064 1.1.1.1.1.1121 915 20930910 22897 256 0.0111805 1.1.1.1.1.1.1.6 916 21138810 5725 64 0.011179 1.1.1.1.1.1129 917 21189630 5729 64 0.0111712 1.1.1.1.1.1132 918 21221970 5730 64 0.0111693 1.1.1.1.1.1134 919 21249690 5733 64 0.0111634 1.1.1.1.1.1135 920 21258930 5735 64 0.0111595 1.1.1.1.1.1136 921 21272790 5736 64 0.0111576 1.1.1.1.1.1137 922 21300510 5738 64 0.0111537 1.1.1.1.1.1138 923 21342090 5739 64 0.0111518 1.1.1.1.1.1140 924 21346710 5741 64 0.0111479 1.1.1.1.1.1141 925 21429870 5743 64 0.011144 1.1.1.1.1.1143 926 21439110 5746 64 0.0111382 1.1.1.1.1.1144 927 21466830 5747 64 0.0111362 1.1.1.1.1.1146 928 21508410 5749 64 0.0111324 1.1.1.1.1.1147 929 21526890 5751 64 0.0111285 1.1.1.1.1.1148 930 21568470 5754 64 0.0111227 1.1.1.1.1.1150 931 21647010 5758 64 0.011115 1.1.1.1.1.1154 932 21660870 5760 64 0.0111111 1.1.1.1.1.1155 933 21693210 5762 64 0.0111073 1.1.1.1.1.1156 934 21744030 5764 64 0.0111034 1.1.1.1.1.1159 935 21785610 5767 64 0.0110976 1.1.1.1.1.1162 936 21790230 5768 64 0.0110957 1.1.1.1.1.1163 937 21854910 5770 64 0.0110919 1.1.1.1.1.1166 938 21882630 5772 64 0.011088 1.1.1.1.1.1169 939 21924210 5774 64 0.0110842 1.1.1.1.1.1171 940 21951930 23110 256 0.0110775 1.1.1.1.1.1.1.7 941 22021230 5780 64 0.0110727 1.1.1.1.1.1175 942 22053570 5783 64 0.0110669 1.1.1.1.1.1177 943 22145970 5788 64 0.0110574 1.1.1.1.1.1179 944 22178310 5789 64 0.0110554 1.1.1.1.1.1180 945 22206030 5790 64 0.0110535 1.1.1.1.1.1181 946 22229130 5792 64 0.0110497 1.1.1.1.1.1183 947 22242990 5794 64 0.0110459 1.1.1.1.1.1184 948 22275330 5795 64 0.011044 1.1.1.1.1.1186 949 22316910 5796 64 0.0110421 1.1.1.1.1.1188 950 22353870 5799 64 0.0110364 1.1.1.1.1.1189 951 22381590 5801 64 0.0110326 1.1.1.1.1.1191 952 22400070 5803 64 0.0110288 1.1.1.1.1.1192 953 22450890 5804 64 0.0110269 1.1.1.1.1.1193 954 22455510 5805 64 0.011025 1.1.1.1.1.1194 955 22483230 5807 64 0.0110212 1.1.1.1.1.1195 956 22497090 5808 64 0.0110193 1.1.1.1.1.1196 957 22506330 5810 64 0.0110155 1.1.1.1.1.1197 958 22520190 5811 64 0.0110136 1.1.1.1.1.1198 959 22561770 5813 64 0.0110098 1.1.1.1.1.1199 960 22594110 5816 64 0.0110041 1.1.1.1.1.1201 961 22644930 5818 64 0.0110003 1.1.1.1.1.1204 962 22677270 5819 64 0.0109985 1.1.1.1.1.1206 963 22704990 5822 64 0.0109928 1.1.1.1.1.1207 964 22714230 5823 64 0.0109909 1.1.1.1.1.1208 965 22755810 5824 64 0.010989 1.1.1.1.1.1210 966 22769670 5825 64 0.0109871 1.1.1.1.1.1211 967 22802010 5826 64 0.0109852 1.1.1.1.1.1213 968 22829730 5829 64 0.0109796 1.1.1.1.1.1214 969 22871310 5831 64 0.0109758 1.1.1.1.1.1216 970 22922130 5832 64 0.0109739 1.1.1.1.1.1218 971 22940610 5833 64 0.0109721 1.1.1.1.1.1220 972 22963710 5835 64 0.0109683 1.1.1.1.1.1221 973 22982190 5836 64 0.0109664 1.1.1.1.1.1222 974 23023770 5837 64 0.0109645 1.1.1.1.1.1223 975 23093070 11684 128 0.0109552 1.1.1.1.1.1.130 976 23116170 5846 64 0.0109477 1.1.1.1.1.1225 977 23185470 5849 64 0.010942 1.1.1.1.1.1227 978 23240910 5853 64 0.0109346 1.1.1.1.1.1229 979 23254770 5854 64 0.0109327 1.1.1.1.1.1230 980 23310210 5858 64 0.0109252 1.1.1.1.1.1233 981 23328690 5859 64 0.0109234 1.1.1.1.1.1235 982 23337930 5860 64 0.0109215 1.1.1.1.1.1236 983 23407230 5862 64 0.0109178 1.1.1.1.1.1238 984 23421090 5863 64 0.0109159 1.1.1.1.1.1239 985 23425710 5864 64 0.0109141 1.1.1.1.1.1240 986 23448810 5865 64 0.0109122 1.1.1.1.1.1241 987 23490390 5866 64 0.0109103 1.1.1.1.1.1244 988 23518110 5867 64 0.0109085 1.1.1.1.1.1246 989 23545830 5871 64 0.010901 1.1.1.1.1.1247 990 23587410 5873 64 0.0108973 1.1.1.1.1.1248 991 23615130 5874 64 0.0108955 1.1.1.1.1.1249 992 23633610 11751 128 0.0108927 1.1.1.1.1.1.132 993 23661330 5877 64 0.0108899 1.1.1.1.1.1250 994 23670570 5878 64 0.0108881 1.1.1.1.1.1251 995 23684430 5879 64 0.0108862 1.1.1.1.1.1252 996 23716770 5881 64 0.0108825 1.1.1.1.1.1254 997 23726010 5882 64 0.0108807 1.1.1.1.1.1255 998 23767590 5883 64 0.0108788 1.1.1.1.1.1257 999 23795310 5885 64 0.0108751 1.1.1.1.1.1258 1000 23823030 5888 64 0.0108696 1.1.1.1.1.1260 1001 23864610 5889 64 0.0108677 1.1.1.1.1.1262 1002 23892330 5890 64 0.0108659 1.1.1.1.1.1265 1003 23924670 5892 64 0.0108622 1.1.1.1.1.1266 1004 23933910 11786 128 0.0108603 1.1.1.1.1.1.133 1005 24003210 5895 64 0.0108567 1.1.1.1.1.1268 1006 24021690 5898 64 0.0108511 1.1.1.1.1.1269 1007 24086370 5901 64 0.0108456 1.1.1.1.1.1270 1008 24146430 5902 64 0.0108438 1.1.1.1.1.1273 1009 24155670 5904 64 0.0108401 1.1.1.1.1.1274 1010 24160290 5905 64 0.0108383 1.1.1.1.1.1275 1011 24201870 5907 64 0.0108346 1.1.1.1.1.1277 1012 24224970 5908 64 0.0108328 1.1.1.1.1.1278 1013 24252690 5909 64 0.0108309 1.1.1.1.1.1279 1014 24257310 5910 64 0.0108291 1.1.1.1.1.1280 1015 24285030 5911 64 0.0108273 1.1.1.1.1.1281 1016 24294270 11832 128 0.0108181 1.1.1.1.1.1.134 1017 24354330 11838 128 0.0108126 1.1.1.1.1.1.135 1018 24460590 5920 64 0.0108108 1.1.1.1.1.1286 1019 24502170 5922 64 0.0108072 1.1.1.1.1.1289 1020 24516030 5924 64 0.0108035 1.1.1.1.1.1290 1021 24548370 5925 64 0.0108017 1.1.1.1.1.1291 1022 24576090 5926 64 0.0107999 1.1.1.1.1.1293 1023 24603810 5929 64 0.0107944 1.1.1.1.1.1294 1024 24617670 5930 64 0.0107926 1.1.1.1.1.1295 1025 24640770 5932 64 0.0107889 1.1.1.1.1.1297 1026 24654630 11875 128 0.0107789 1.1.1.1.1.1.136 1027 24714690 11881 128 0.0107735 1.1.1.1.1.1.137 1028 24807090 5943 64 0.010769 1.1.1.1.1.1305 1029 24834810 11894 128 0.0107617 1.1.1.1.1.1.138 1030 24894870 11901 128 0.0107554 1.1.1.1.1.1.139 1031 25019610 5953 64 0.0107509 1.1.1.1.1.1311 1032 25056570 5954 64 0.0107491 1.1.1.1.1.1313 1033 25070430 5955 64 0.0107473 1.1.1.1.1.1314 1034 25088910 5956 64 0.0107455 1.1.1.1.1.1316 1035 25102770 5957 64 0.0107437 1.1.1.1.1.1317 1036 25139730 5958 64 0.0107419 1.1.1.1.1.1318 1037 25153590 5959 64 0.0107401 1.1.1.1.1.1319 1038 25158210 5960 64 0.0107383 1.1.1.1.1.1320 1039 25185930 5961 64 0.0107365 1.1.1.1.1.1321 1040 25195170 11938 128 0.0107221 1.1.1.1.1.1.140 1041 25347630 5971 64 0.0107185 1.1.1.1.1.1327 1042 25361490 5972 64 0.0107167 1.1.1.1.1.1328 1043 25416930 5976 64 0.0107095 1.1.1.1.1.1331 1044 25472370 5977 64 0.0107077 1.1.1.1.1.1332 1045 25518570 5979 64 0.0107041 1.1.1.1.1.1333 1046 25569390 5982 64 0.0106988 1.1.1.1.1.1336 1047 25601730 5983 64 0.010697 1.1.1.1.1.1338 1048 25615590 11987 128 0.0106782 1.1.1.1.1.1.141 1049 25735710 12001 128 0.0106658 1.1.1.1.1.1.142 1050 25795770 12006 128 0.0106613 1.1.1.1.1.1.143 1051 25915890 12018 128 0.0106507 1.1.1.1.1.1.144 1052 26100690 6011 64 0.0106471 1.1.1.1.1.1361 1053 26128410 6012 64 0.0106454 1.1.1.1.1.1362 1054 26142270 6013 64 0.0106436 1.1.1.1.1.1363 1055 26151510 6014 64 0.0106418 1.1.1.1.1.1364 1056 26220810 6017 64 0.0106365 1.1.1.1.1.1366 1057 26262390 6021 64 0.0106295 1.1.1.1.1.1368 1058 26336310 12064 128 0.0106101 1.1.1.1.1.1.145 1059 26456430 12081 128 0.0105951 1.1.1.1.1.1.146 1060 26516490 12089 128 0.0105881 1.1.1.1.1.1.147 1061 26636610 12105 128 0.0105741 1.1.1.1.1.1.148 1062 26927670 6053 64 0.0105733 1.1.1.1.1.1395 1063 26973870 6056 64 0.010568 1.1.1.1.1.1396 1064 27001590 6057 64 0.0105663 1.1.1.1.1.1398 1065 27024690 6059 64 0.0105628 1.1.1.1.1.1399 1066 27066270 6063 64 0.0105558 1.1.1.1.1.1401 1067 27098610 6064 64 0.0105541 1.1.1.1.1.1403 1068 27126330 6065 64 0.0105523 1.1.1.1.1.1404 1069 27204870 6068 64 0.0105471 1.1.1.1.1.1405 1070 27218730 6069 64 0.0105454 1.1.1.1.1.1407 1071 27232590 6071 64 0.0105419 1.1.1.1.1.1408 1072 27237210 12166 128 0.0105211 1.1.1.1.1.1.149 1073 27357330 12179 128 0.0105099 1.1.1.1.1.1.150 1074 27597570 12207 128 0.0104858 1.1.1.1.1.1.151 1075 27897870 12240 128 0.0104575 1.1.1.1.1.1.152 1076 28138110 12266 128 0.0104353 1.1.1.1.1.1.153 1077 28258230 12276 128 0.0104268 1.1.1.1.1.1.154 1078 28438410 12294 128 0.0104116 1.1.1.1.1.1.155 1079 28618590 12314 128 0.0103947 1.1.1.1.1.1.156 1080 29034390 6158 64 0.010393 1.1.1.1.1.1496 1081 29039010 12359 128 0.0103568 1.1.1.1.1.1.157 1082 29159130 12373 128 0.0103451 1.1.1.1.1.1.158 1083 29339310 12392 128 0.0103292 1.1.1.1.1.1.159 1084 29519490 12412 128 0.0103126 1.1.1.1.1.1.160 1085 29759730 12439 128 0.0102902 1.1.1.1.1.1.161 1086 29939910 12455 128 0.010277 1.1.1.1.1.1.162 1087 30300270 12494 128 0.0102449 1.1.1.1.1.1.163 1088 30420390 12505 128 0.0102359 1.1.1.1.1.1.164 1089 30600570 12520 128 0.0102236 1.1.1.1.1.1.165 1090 30660630 12526 128 0.0102187 1.1.1.1.1.1.166 1091 30960930 12557 128 0.0101935 1.1.1.1.1.1.167 1092 31020990 12563 128 0.0101886 1.1.1.1.1.1.168 1093 31201170 12580 128 0.0101749 1.1.1.1.1.1.169 1094 31501470 12612 128 0.0101491 1.1.1.1.1.1.170 1095 31561530 12617 128 0.010145 1.1.1.1.1.1.171 1096 31861830 12643 128 0.0101242 1.1.1.1.1.1.172 1097 31921890 12649 128 0.0101194 1.1.1.1.1.1.173 1098 32102070 12668 128 0.0101042 1.1.1.1.1.1.174 1099 32642610 12719 128 0.0100637 1.1.1.1.1.1.175 1100 32762730 12730 128 0.010055 1.1.1.1.1.1.176 1101 32822790 12737 128 0.0100495 1.1.1.1.1.1.177 1102 32942910 12748 128 0.0100408 1.1.1.1.1.1.178 1103 33123090 12764 128 0.0100282 1.1.1.1.1.1.179 1104 33303270 12783 128 0.0100133 1.1.1.1.1.1.180 1105 33543510 12803 128 0.00999766 1.1.1.1.1.1.181 1106 33723690 12819 128 0.00998518 1.1.1.1.1.1.182 1107 33903870 12839 128 0.00996962 1.1.1.1.1.1.183 1108 34564530 12899 128 0.00992325 1.1.1.1.1.1.184 1109 34624590 12907 128 0.0099171 1.1.1.1.1.1.185 1110 34924890 12928 128 0.00990099 1.1.1.1.1.1.186 1111 35165130 12957 128 0.00987883 1.1.1.1.1.1.187 1112 35465430 12981 128 0.00986057 1.1.1.1.1.1.188 1113 35645610 12996 128 0.00984918 1.1.1.1.1.1.189 1114 35825790 13012 128 0.00983707 1.1.1.1.1.1.190 ////// Observations ////// 1. There are 2 non-squarefree terms {18, 150} less than 36,000,000. 2. No primes p appear in b(i). This is because all regular m divide p, and since all the regulars of 1 also divide 1, no primes appear in b(i). If we were to ignore 1, then the prime 2 would set a record for the ratio of rcf(k)/tau(k). 3. The values of tau(b(n)) are in A000079, i.e., powers 2^e except e = 1 (see number 2 above). 4. Aside from the 2 non-squarefree terms, many terms m are products of A002110(j) * p_h, with h > j between some lower and upper bound outside of when m is in A002110. Example: 30 is in A002110; {42, 66, 78, 102, 114, 138, 174} are A002110(3) * p_h with 2 <= h <= 8. 5. There are a few terms of the form A002110(j) * p_g * p_h, with g > j + 1 and h > g. In other words, there is a gap in the indices of the prime divisors between the 3rd and 2nd largest prime divisors, as well as one potentially between the 2nd and largest prime divisors. The smallest m of this type is 46410 = 2 * 3 * 5 * 7 * 13 * 17, followed by 51870 = 2 * 3 * 5 * 7 * 13 * 19. 6. Regarding A001221(b(i)), the sequence seems to be divided into "tiers" wherein omega(b(i)) either increases or holds as i increases. Thus we might speak of "tiers" for all terms m <= A002110(6) = 30030. However, 36330 immediately follows 30030 in the sequence and omega(36330) = 5. For m > 30030, the numbers m with omega of a particular value are increasingly mixed with terms that have a different value. These intermingled "tiers" do seem to have a floor and ceiling. The tiers consist of terms as described in number 3 above: Tier 0 contains {1}. There is no tier 1 since this would contain numbers m with omega(m) = 1. Tier 2 contains {6, 10, 18, 22}; 12 is missing since rcf(12)/tau(12) = 8/6 was bested by 10 with ratio 6/4. Otherwise, A002110(1) * h with 1 <= h <= 4 and h =/= 3, also accepting the term 18. Tier 3 contains A002110(2) * h with 1 <= h <= 8, as well as 150 = 2 * 3 * 5^2. Tier 4 contains A002110(3) * h with 1 <= h <= 18, and h =/= 8. Tier 5 contains A002110(4) * h with 1 <= h <= 30. This tier is interrupted by A002110(6) as before mentioned, but continues after 30030 with 36 <= h <= 38. There are further intrusions of numbers with omega(m) = 6, as well as missing values in the range 47 <= h <= 356. The 5th tier seems to terminate at b(319) = A002110(7), whereinafter no number with omega(5) appears. "Tier" 6 has a floor at A002110(6) and its ceiling has not yet been seen in computation. 7. "Messy bands" of terms m in A244052 appear in b(i). The indices j of these terms m in A244052 appear as follows, usually with significant intrusions of b(i) not in A244052: A244052 tier 0: j = 1 at i = 1. A244052 tier 2: j = {4, 5, 7} starting at b(3) = 6 = A002110(2); includes the non-squarefree 18. A244052 tier 3: j = {9, 10, 15} starting at b(6) = 30 = A002110(3); includes the non-squarefree 150. A244052 tier 4: 17 <= j <= 19 at 15 <= i <= 17 respectively. (b(15) = 210 = A002110(4)). A244052 tier 5: 29 <= j <= 32, at 32 <= i <= 35 respectively. (b(32) = 2310 = A002110(5)). A244052 tier 6: 48 <= j <= 53 starting at b(62) = 30030 = A002110(6). A244052 tier 7: 72 <= j <= 77 starting at b(319) = 510510 = A002110(7). A244052 tier 7: 104 <= j <= 111* starting at b(777) = 9699690 = A002110(8). * it is unclear whether this is the maximum value of j for m in A244052 tier 7. ////// Conjectures /////// 1. The only non-squarefree terms in b(i) are 18 and 150. 2. Primorials A002110(j) for j = 0 and j > 2 appear in b(i). 3. Aside from 18 and 150, there are a few of the smallest terms k in tier t of A244052 in b(i). These terms are squarefree and also appear in A288813. 4. There will always be "intrusions" of m of omega(m) < t among terms k in tier t of A244052 with t > 6, equating to m >= A002110(6) i.e., m >= 30030. Corollary to this is that the ranges j = 1, 17 <= j <= 19 and 29 <= j <= 32 in A244052 are the only contiguous terms of A244052 tier t in b(i). +---------------------------------------------------+ | Table 3: The intersection of A002182 and A244052. | +---------------------------------------------------+ Based on Conjecture 1.1, we re-examine the terms m that are in both A002182 and A244052, then produce a chart plotting them with n = primorial A002110(omega(m)) and k = m/A002110(omega(m)). The intersection of A002182 and A244052 is finite, consisting of 13 terms: {1, 2, 4, 6, 12, 24, 60, 120, 180, 840, 1260, 1680, 27720}. All of these terms are also in A060735 and not in A288813, as the latter are squarefree and have "gaps" among prime divisors. This intersection has the following number of terms in the "tiers" 0 through 5 of A244052: {1, 2, 3, 3, 3, 1}. A "tier" n consists of all terms m in A244052 with A002110(n) <= m < A002110(n + 1). If we look at A060735 as a number triangle T(n,k) = k * A002110(n) with 1 <= k < prime(n + 1), the terms are plotted below. The top figure is the coordinates (n,k) in A060735. The middle figure is the decimal number, and the bottom set of numbers delimited by "." are the exponents of the prime pertaining to the place in which the exponent appears. For example, "2.1.1" -> 2^2 * 3^1 * 5^1 = 60. k 1 2 3 4 5 6 7 8 9 10 11 12 +-------------------------------------------------------------------------- A002110(0) | {0,1} | 1 | 0 | A002110(1) | {1,1} {1,2} | 2 4 | 1 2 | A002110(2) | {2,1} {2,2} . {2,4} | 6 12 24 n | 1.1 2.1 3.1 | A002110(3) | . {3,2} . {3,4} . {3,6} | 60 120 180 | 2.1.1 3.1.1 2.2.1 | A002110(4) | . . . {4,4} . {4,6} . {4,8} . . | 840 1260 1680 | 3.1.1.1 2.2.1.1 4.1.1.1 | A002110(5) | . . . . . . . . . . . {5,12} | 27720 | 3.2.1.1.1 | ////// Observations ////// 1. The terms in the multiplicity notations of these numbers hold or decrease. For k = 3 or k = 5, we would have 1.2, 1.2.1, 1.2.1.1, or 1.1.2, 1.1.2.1, 1.1.2.1.1, etc. Such configurations of prime divisors are absent from terms in A002182. 2. If we were to extend the irregular triangle beyond k < p_(n + 1), then we could perhaps plot all terms of A002182 on a grid. In the grid below, dots "." appear for terms in A060735, if not occupied by a term in A002182, which if in A060735 is followed by an asterisk. These dotted and asterisked positions are fully occupied by many but not all terms in A244052 and illustrates some of the "divergence" of the two record-setter sequences. 0 1 2 3 4 5 6 7 8 9 10 +------------------------------------------------------------ 1 | 1* 2* 6* . . . . . . . . 2 | 4* 12* 60* . . . . . . . 3 | . . . . . . . . . 4 | 24* 120* 840* . . . . . . 5 | . . . . . . . . 6 | 36 180* 1260* . . . . . . 7 | . . . . . . . 8 | 48 240 1680* . . . . . . 9 | . . . . . . . 10 | . . . . . . . 11 | . . . . . . 12 | 360 2520 27720* . . . . . 13 | . . . . 14 | . . . . 15 | . . . . 16 | . . . . 17 | . . . 18 | . . . 19 | . . 20 | . . 21 | . . 22 | . . 23 | . 24 | 720 5040 55440 720720 . 3. Since we observe the terms plotted on a grid tend to occupy certain values of k, we might condense the grid and merely annotate coordinates for each of the terms. We can use {omega(m),m/A002110(omega(m))} to furnish T(n,k). +---------------------------------------------------------------------+ | Table 4: Plot {A108602(m), m/A002110(A108602(m))} for m in A002182. | +---------------------------------------------------------------------+ This is a condensed table similar to that of Observations 3.2 and 3.3. The coordinates {n,k} may have k further abbreviated to multiplicity notation, but are presented decimally here. {0,1} {1,1} {2,1} 1 2* 6* {1,2} {2,2} {3,2} 4 12* 60* {2,4} {3,4} {4,4} 24 120* 840 {2,6} {3,6} {4,6} 36 180 1260 {2,8} {3,8} {4,8} 48 240 1680 {3,12} {4,12} {5,12} 360* 2520* 27720 {3,24} {4,24} {5,24} {6,24} 720 5040* 55440* 720720* {4,36} {5,36} {6,36} 7560 83160 1081080 {4,48} {5,48} {6,48} 10080 110880 1441440* {4,72} {5,72} {6,72} {7,72} {8,72} 15120 166320 2162160 36756720 698377680 {4,96} {5,96} {6,96} 20160 221760 2882880 {4,120} {5,120} {6,120} {7,120} 25200 277200 3603600 61261200 {5,144} {6,144} {7,144} {8,144} 332640 4324320* 73513440 1396755360 {4,216} {5,216} {6,216} {7,216} {8,216} 45360 498960 6486480 110270160 2095133040 {4,240} {4,240} {6,240} {7,240} {8,240} 50400 554400 7207200 122522400 2327925600 {5,288} {6,288} {7,288} {8,288} {9,288} 665280 8648640 147026880 2793510720 64250746560 {6,360} {7,360} {8,360} {9,360} 10810800 183783600 3491888400 80313433200 {6,480} {7,480} {8,480} 14414400 245044800 4655851200 {6,576} {7,576} {8,576} {9,576} 17297280 294053760 5587021440 128501493120 {6,720} {7,720} {8,720} {9,720} 21621600* 367567200* 6983776800* 160626866400 {6,1080} {7,1080} {8,1080} {9,1080} 32432400 551350800 10475665200 240940299600 {6,1440} {7,1440} {8,1440} {9,1440} . 43243200 735134400 13967553600* 321253732800* . {7,2160} {8,2160} {9,2160} 1102701600 20951330400 481880599200 {8,2880} {9,2880} . 27935107200 642507465600 . {7,4320} {8,4320} {9,4320} . 2205403200 41902660800 963761198400 . {8,5040} {9,5040} . 48886437600 1124388064800 . {9,7200} . 1606268664000 . {8,7560} {9,7560} . 73329656400 1686582097200 . {9,8640} . 1927522396800 . {8,10080} {9,10080} . 97772875200 2248776129600 . {9,14400} . 3212537328000 . {8,15120} {9,15120} . 146659312800 3373164194400 . {9,20160} . 4497552259200 . {8,30240} {9,30240} . 293318625600 6746328388800 . {9,40320} . 8995104518400 . {9,60480} . 13492656777600 . {9,120960} . 26985313555200 . ////// Observations ////// 1. The k-values appear to occupy "streets" of values: {1, 2, 4, 6, 8, 12, 24, 36, 48, 72, 96, 120, 144, 216, 240, 288, 360, 480, 576, 720, 1080, 1440, 2160, 2880, ...}. 2. The chart resembles the periodic table, except that the range of valid values evacuate the smaller k-values as n increases. Some interposing k-values appear for (n + 1) while other k-values valid for (n - 1) prove invalid in column n, even though they are flanked above and below by valid k-values. 3. k = 72 is curiously "strong", valid for 4 <= n <= 8. This seems true also for k = 1440. 4. Terms in A002201 seem to appear in clusters: {1,1}, {2,1}, {2,2}, {3,2}, {3,4}, {3,12}, {4,12}, {4,24}, {5,24}, {6,24}, {6,48}, {6,144}, {6,720}, {7,720}, {8,720}, {9,1440}, ... +----------------------------------------------------------------------------+ | Graph 5: Plot "x" at {A108602(m), m/A002110(A108602(m))} for m in A002182. | +----------------------------------------------------------------------------+ Mathematica output involving the smallest 10,000 HCNs obtained at [Flammenkamp 1]. We place an "x" at the location of an HCN according to the coordinates {A108602(m), m/A002110(A108602(m))}. We place an "0" at the location of a SHCN (in A002201). This is a compressed version of Tables 3 and 4. This table is clipped where the families' ranges appear to be exhausted. A larger table was produced using 2^16 terms of [Flammenkamp 1] but is too large to be reproduced here. 111111111122222222223333333333 Family 0123456789012345678901234567890123456789 ---------------------------------------------------------------- 0 x00 1 x00 2 x0x 1.1 xxx 3 xxx 2.1 00x 3.1 x000 2.2 xxx 4.1 xx0 3.2 xxxxx 5.1 xxx 3.1.1 xxxx 4.2 x0xx 3.3 xxxxx 4.1.1 xxxxx 5.2 xxxxx 3.2.1 xxxx 5.1.1 xxx 6.2 xxxx 4.2.1 000x 3.3.1 xxxx 5.2.1 xx00xx 4.3.1 xxx 6.2.1 xxxx 5.3.1 xxxxx 4.2.1.1 xxxx 5.2.2 xxx 3.3.1.1 xxxx 6.3.1 xxx 5.2.1.1 x00xxxx 6.2.2 xxxxx 4.3.1.1 xxxxx 6.2.1.1 xxxxxx 5.3.1.1 xx00xxxxx 7.2.1.1 xxxxxx 5.2.2.1 xxxxx 6.3.1.1 xx00000x 5.4.1.1 xxxxxx 6.2.2.1 xxxxxxxx 7.3.1.1 xxxxxxxx 5.3.2.1 xxxxxxxx 6.4.1.1 xxxxxx 8.3.1.1 xxxxxx 6.3.2.1 xxxx000x 7.4.1.1 xxxxxx 5.4.2.1 xxxxx 7.3.2.1 xxxxx0x 6.4.2.1 xxxxxxxx 8.3.2.1 xxxxxxxxx 7.3.1.1.1 xxxxx 5.3.2.1.1 xxxxx 7.4.2.1 xxxxxxx 9.3.2.1 xxxxxx 8.3.1.1.1 xxxxx 6.3.2.1.1 xxxxxx 7.4.1.1.1 xxxxx 5.4.2.1.1 xxxxxx 6.4.2.2 xxxxx 7.3.2.1.1 x00xxx 6.4.2.1.1 xxxxxxxxx 8.3.2.1.1 xxxxxxxxx 7.3.1.1.1.1 xxxxxx 7.4.2.1.1 xx00000xx 9.3.2.1.1 xxxxxxxxx 6.5.2.1.1 xxxxxx 7.3.3.1.1 xxxxxx 8.4.2.1.1 xxxxxxx 7.4.1.1.1.1 xxxxxxxx 7.5.2.1.1 xxxxxxx 6.4.2.2.1 xxxxxxx 9.4.2.1.1 xxxxxxx 7.3.2.1.1.1 xxxxxxxxx 8.5.2.1.1 xxxxx 6.4.2.1.1.1 xxxxxxxxxx 8.3.2.1.1.1 xxxxxxxxxx 9.5.2.1.1 xxxxx 7.4.2.1.1.1 xxxx000xxxxxx 9.3.2.1.1.1 xxxxxxxxx 6.5.2.1.1.1 xxxxxxxxx 7.3.3.1.1.1 xxxxxxxx 8.4.2.1.1.1 xxx00000xx 7.5.2.1.1.1 xxxxxxxxxx 6.4.2.2.1.1 xxxxxxx 9.4.2.1.1.1 xxxxxxxxxx 8.3.2.2.1.1 xxxxxxxx 7.4.3.1.1.1 xxxxxxxxxx 8.5.2.1.1.1 xxxxxxxxxx 7.4.2.2.1.1 xxxxxxxxx 9.3.2.2.1.1 xxxxxxxx 8.4.3.1.1.1 xxxxxxxxxx 9.5.2.1.1.1 xxxxxxxxxx 8.4.2.2.1.1 xxxx000xxxxxx 7.4.2.1.1.1.1 xxxxxxxxxx 9.4.3.1.1.1 xxxxxxxx 7.5.2.2.1.1 xxxxxxxxx 9.4.2.2.1.1 xxxxxxxxxxxxx 8.4.2.1.1.1.1 xxxxxxx 8.5.2.2.1.1 xxxxxxxxxxxxxx 7.5.2.1.1.1.1 xxxxxxx 10.4.2.2.1.1 xxx 9.5.3.1.1.1 xxxxxxx 9.4.2.1.1.1.1 xxxxxxx 8.4.3.2.1.1 xxxxxxxxxx 9.5.2.2.1.1 xxxxxxxxxxxxxx 8.5.2.1.1.1.1 xxxxxxx 7.4.2.2.1.1.1 xxxxxxx 9.4.3.2.1.1 xxxxxxxxxx 10.5.2.2.1.1 xxxx 8.4.3.1.1.1.1 xxxxxxx 9.5.2.1.1.1.1 xxxxxxx 8.5.3.2.1.1 xxxxxxxxxxx 8.4.2.2.1.1.1 00000xxxxx 7.4.2.1.1.1.1.1 xxxxxxx 9.4.3.1.1.1.1 xxx 7.5.2.2.1.1.1 xxxxxxx 9.5.3.2.1.1 xxxxxxxxxxx 9.4.2.2.1.1.1 xxxxxxxxxx 7.4.3.2.1.1.1 xxxxxxx 8.5.2.2.1.1.1 xxxx0xxxxxxxx 10.4.2.2.1.1.1 xxx 9.5.3.1.1.1.1 xxxx 8.4.3.2.1.1.1 xxxxxxxxxx 9.5.2.2.1.1.1 xxxxxxxxxxxxx 7.5.3.2.1.1.1 xxxxxx 8.6.2.2.1.1.1 xxxxxxx 7.4.2.2.1.1.1.1 xxxxxxxxxx 9.4.3.2.1.1.1 xxxxxxxxxx 10.5.2.2.1.1.1 xxxxxxxxxxx 8.4.3.1.1.1.1.1 xxxxxxxx 8.5.3.2.1.1.1 xxx000xxxxxxxxxx 9.6.2.2.1.1.1 xxxxxxx 8.4.2.2.1.1.1.1 xxxxxxxxxx 11.5.2.2.1.1.1 xxxxxxxxxxx 9.4.3.1.1.1.1.1 xx 7.5.2.2.1.1.1.1 xxxxxxxxxxx 9.5.3.2.1.1.1 xxxxxxxxxxxxxxxxx 9.4.2.2.1.1.1.1 xxxxxxxxxxxxxxx 8.6.3.2.1.1.1 xxxxxxxxx 7.4.3.2.1.1.1.1 xxxxxxxxxxx 8.5.2.2.1.1.1.1 xxxxxxxxxxxxxxxx 10.4.2.2.1.1.1.1 xxxx 9.5.3.1.1.1.1.1 xxxxx 9.6.3.2.1.1.1 xxxxxxxxxxx 8.4.3.2.1.1.1.1 xxxxxxxxxxxxxxx 9.5.2.2.1.1.1.1 xxxxxxxxxxxxxxxxxxx 7.5.3.2.1.1.1.1 xxxxxxxxx 8.6.2.2.1.1.1.1 xxxxxxxxx 9.4.3.2.1.1.1.1 xxxxxxxxxxxxxxx 10.5.2.2.1.1.1.1 xxxxxxxxxxxxxxx 8.5.3.2.1.1.1.1 xx0xxxxxxxxxxxxxxxxxx 9.6.2.2.1.1.1.1 xxxxxxxxxxxx 10.4.3.2.1.1.1.1 xxxxxxxxxxx 11.5.2.2.1.1.1.1 xxxxxxxxxxxxxxx 9.5.3.2.1.1.1.1 xxxx000000000000xxxxxxx 11.4.3.2.1.1.1.1 xxxxxxxxxxx 8.6.3.2.1.1.1.1 xxxxxxxxxxxxxxxx 7.4.3.2.1.1.1.1.1 xxxxxx 10.5.3.2.1.1.1.1 xxxxxxxxxxxxxxxxxxx 9.6.3.2.1.1.1.1 xxxxxxxxxxxxxxxx 8.4.3.2.1.1.1.1.1 xxxxxxxxxxxxxxxxxx 11.5.3.2.1.1.1.1 xxxxxxxxxxxxxxxxxxx 9.5.2.2.1.1.1.1.1 xxxxxxxxxxxxxxxxxxx 8.5.3.2.2.1.1.1 xxxxxxxx 7.5.3.2.1.1.1.1.1 xxxxxxxxxxxx 10.6.3.2.1.1.1.1 xxxxxxxxxxxxxx 9.5.3.3.1.1.1.1 xxxxxxxxxxxxxxxxx 9.4.3.2.1.1.1.1.1 xxxxxxxxxxxxxxxxxxx 10.5.2.2.1.1.1.1.1 xxxxxxxxxxxxxxxxx 9.5.3.2.2.1.1.1 xxxxxxxxxxxxx 8.5.3.2.1.1.1.1.1 xxxxxxxxxxxxxxxxxxxxxxxxx 9.6.2.2.1.1.1.1.1 xxxxxxxxxxxxx 10.4.3.2.1.1.1.1.1 xxxxxxxxxxxxxx 11.5.2.2.1.1.1.1.1 xxxxxxxxxxxxxxxxx 9.6.3.3.1.1.1.1 xxxxxxxxx 10.5.3.2.2.1.1.1 xxxxxxxxxxxx 9.5.3.2.1.1.1.1.1 xxxxxxxxxx0000000xxxxxxxxxxxxx 11.4.3.2.1.1.1.1.1 xxxxxxxxxxxx 9.6.3.2.2.1.1.1 xxxxxxxxxxxxxxx 8.6.3.2.1.1.1.1.1 xxxxxxxxxxxxxxxxxx 11.5.3.2.2.1.1.1 xxxxxxxxxxxx 10.5.3.2.1.1.1.1.1 xxxxxxxxxxx0000000000xxxx 10.6.3.2.2.1.1.1 xxxxxxxxxxxxxxxx 9.6.3.2.1.1.1.1.1 xxxxxxxxxxxxxxxxxxxxx 11.5.3.2.1.1.1.1.1 xxxxxxxxxxxxxxxxxxxxxxxxx 9.5.4.2.1.1.1.1.1 xxxxxxxxxxxxxxxxx 8.5.3.2.2.1.1.1.1 xxxxxxxxxxxx 10.6.3.2.1.1.1.1.1 xxxxxxxxxxxxxxxx0000x 9.5.3.3.1.1.1.1.1 xxxxxxxxxxxxxxxxxxxxx 12.5.3.2.1.1.1.1.1 xxxxxxxxxxxxxxxxxxxxxxxxx 9.7.3.2.1.1.1.1.1 xxxxxxxxxxxxxxxxx 10.5.4.2.1.1.1.1.1 xxxxxxxxxxxxxxxxx 9.5.3.2.2.1.1.1.1 xxxxxxxxxxxxxxxx 11.6.3.2.1.1.1.1.1 xxxxxxxxxxxxxxxxxxxxx 10.5.3.3.1.1.1.1.1 xxxxxxxxxxxxxxxxxxxx 9.6.4.2.1.1.1.1.1 xxxxxxxxxxxxxxxxx 8.6.3.2.2.1.1.1.1 xxxxxxxxxxxxx 10.7.3.2.1.1.1.1.1 xxxxxxxxxxxxxxxx 11.5.4.2.1.1.1.1.1 xxxxxxxxxxxxxxxxx 9.6.3.3.1.1.1.1.1 xxxxxxxxxxxxxx 10.5.3.2.2.1.1.1.1 xxxxxxxxxxxxxxxx 11.5.3.3.1.1.1.1.1 xxxxxxxxxxxxxxxxxxxx 9.5.3.2.1.1.1.1.1.1 xxxxxxxxxxxxxxxxxxxxxx 9.6.3.2.2.1.1.1.1 xxxxxxxxxxxxxxxxxxxxx 12.5.4.2.1.1.1.1.1 xxxxxxxxxxxxxx 10.6.3.3.1.1.1.1.1 xxxxxxxxxxxxx 11.5.3.2.2.1.1.1.1 xxxxxxxxxxxxxxxxxxxx 9.5.4.2.2.1.1.1.1 xxxxxxxxxxx 10.5.3.2.1.1.1.1.1.1 xxxxxxxxxxxxxxxxxxx 10.6.3.2.2.1.1.1.1 xxxxxxxxxxxxxxxxxxxxx 9.5.3.3.2.1.1.1.1 xxxxxxxxxxxxxxxx 11.6.3.3.1.1.1.1.1 xxxxxxxxxxxxx 9.6.3.2.1.1.1.1.1.1 xxxxxxxxxxxxxx 10.5.4.2.2.1.1.1.1 xxxxxxxxxxx 11.5.3.2.1.1.1.1.1.1 xxxxxxxxxxxxxxxx 11.6.3.2.2.1.1.1.1 xxxxxxxxxxxxxxxxxxx 10.5.3.3.2.1.1.1.1 xxxxxxx 9.6.4.2.2.1.1.1.1 xxxxxxxxxxxxxxx 10.6.3.2.1.1.1.1.1.1 xxxxxxxxxx00xx 9.5.3.3.1.1.1.1.1.1 xxxxxxxxxxxxxxx 11.5.4.2.2.1.1.1.1 xxxxxxxxxxx 9.6.3.3.2.1.1.1.1 xxxxxxxxxxxxxxx 12.5.3.2.1.1.1.1.1.1 xxxxxxxxxxxxxxxx 9.7.3.2.1.1.1.1.1.1 xxxxxxxxxxx 12.6.3.2.2.1.1.1.1 xxxxxxxxxxxxxxxxx 10.5.4.2.1.1.1.1.1.1 xxxxxxxx 11.5.3.3.2.1.1.1.1 xxxxxxxxxxx 9.5.3.2.2.1.1.1.1.1 xxxxxxxx 11.6.3.2.1.1.1.1.1.1 xxxxxxxxxxxxxx 10.5.3.3.1.1.1.1.1.1 xxxxxxxxxxxx 9.6.4.2.1.1.1.1.1.1 xxxxxxxx 10.6.3.3.2.1.1.1.1 xxxxxxxxxxxx 10.7.3.2.1.1.1.1.1.1 xxxxxxxxxxxxxx 11.5.4.2.1.1.1.1.1.1 xxxxxxxx 9.6.3.3.1.1.1.1.1.1 xxxxxxx 10.5.3.2.2.1.1.1.1.1 xxxxxxxx 11.5.3.3.1.1.1.1.1.1 xxxxxxxxx 10.6.3.2.2.2.1.1.1 xxxxxxx 10.6.4.2.1.1.1.1.1.1 xxxxxxxx 9.5.3.2.1.1.1.1.1.1.1 xxxxxxxxxxxxxxx 11.6.3.3.2.1.1.1.1 xxxxxxxxxxxx 9.6.3.2.2.1.1.1.1.1 xxxxxxxxxxxxxx 12.5.4.2.1.1.1.1.1.1 xxxxxxxx 10.6.3.3.1.1.1.1.1.1 xxxxxxx 11.5.3.2.2.1.1.1.1.1 xxxxxxxxxxxxxx 9.5.4.2.2.1.1.1.1.1 xxxxxxxx 11.6.3.2.2.2.1.1.1 xxxxx 11.6.4.2.1.1.1.1.1.1 xxxxxxxx 10.5.3.2.1.1.1.1.1.1.1 xxxxxxxxxxxxxxx 10.6.3.2.2.1.1.1.1.1 000000xxxxxxxx 9.5.3.3.2.1.1.1.1.1 xxxxxxxx 11.6.3.3.1.1.1.1.1.1 xxxxxxx 12.5.3.2.2.1.1.1.1.1 xxxxxxxxxxxxxx 9.6.3.2.1.1.1.1.1.1.1 xxxxxxxxxxxxxxx 10.5.4.2.2.1.1.1.1.1 xxxxxxxx 11.5.3.2.1.1.1.1.1.1.1 xxxxxxxxxxx 11.6.3.2.2.1.1.1.1.1 xxxxxxxxxxxxxx 10.5.3.3.2.1.1.1.1.1 xxxxxxxx 9.6.4.2.2.1.1.1.1.1 xxxxxxxxxxxxxx 10.6.3.2.1.1.1.1.1.1.1 xxxxxxxxxxxxxxx 9.5.3.3.1.1.1.1.1.1.1 xxxxxxxxxxxxx 11.5.4.2.2.1.1.1.1.1 xxxxxxxxxxxx 9.6.3.3.2.1.1.1.1.1 xxxxxxxxxxxxxx 12.5.3.2.1.1.1.1.1.1.1 xxxxxxxxxxx 12.6.3.2.2.1.1.1.1.1 xxxxxxxxxxx 11.5.3.3.2.1.1.1.1.1 xxxxxxxxxxxxxx 10.6.4.2.2.1.1.1.1.1 xxxxxxxxxxxxxx 9.5.3.2.2.1.1.1.1.1.1 xxxxxxxxxxxxxx 11.6.3.2.1.1.1.1.1.1.1 xxxxxxxxxxx 9.6.3.2.2.2.1.1.1.1 xxxxx 10.5.3.3.1.1.1.1.1.1.1 xxxxxxxxxxxxxxxx 10.6.3.3.2.1.1.1.1.1 xxxxxxxxxxxxxx 10.7.3.2.1.1.1.1.1.1.1 xxxxxxxxxxxxxxx 12.5.3.3.2.1.1.1.1.1 xxxxxxxxxxxxxx 11.5.4.2.1.1.1.1.1.1.1 xxxxxxxxxxxxxx 9.6.3.3.1.1.1.1.1.1.1 xxxxxxxxxxxxxx 11.6.4.2.2.1.1.1.1.1 xxxxxxxxxxxxxx 10.5.3.2.2.1.1.1.1.1.1 xxxxxxxxxxxxxx 10.6.3.2.2.2.1.1.1.1 xxxxxxxxxxx 11.6.3.3.2.1.1.1.1.1 xxxxxxxxxxxxxx ------------------------------------------------------------------------------------------------------------- 111111111122222222223333333333444444444455555555556666666666777777777788888 Family 0123456789012345678901234567890123456789012345678901234567890123456789012345678901234 ////// Observations ////// 1. Interesting "strong" families such as 3.2, 5.2.1, 5.2.1.1, 5.3.1.1.1, etc. 2. Interesting "choke points" at 9: 4.3.1, n = {29, 30} family 10.4.2.2.1.1, and more severely, 32: 10.4.2.2.1.1.1 and 36: 8.6.2.2.1.1.1. 3. The "energy" seems to transfer around 18:6.4.2.2 such that for n < 18 the HCNs are mostly and solidly in families k < 6.4.2.2 -> 6350400, but for n > 18, the HCNs are mostly and solidly in families k > 6350400. 4. In using the first 2^16 terms of Achim Flammenkamp's "HCN_NO" data, an expanded chart reveals an actual "disjunction" between families 14.9.4.4.2.2.2.2.1.1.1.1.1.1.1.1.1.1.1.1 and 14.8.4.3.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.1, where the former does not produce HCNs for 5 tiers until the latter begins to produce HCNs. 5. The "amplitude" of the families (meaning the start and end of rows k) seems to increase and dominate as the family k increases. 6. Superior highly composite numbers appear in "rows" at particular families, for example, the five SHCNs at tiers 11 through 15 inclusive at family 6.3.1.1. (i.e., the 22nd through 25th terms in A002201, specifically, 12129898443062400, 448806242393308800, 18401055938125660800, 791245405339403414400, 37188534050951960476800). +--------------------------------------------------+ | Table 6: m/A002110(A108602(m)) for m in A002182. | +--------------------------------------------------+ k = m/A002110(A108602(m)) for m in A002182. i = index. pop. = length of row k in Graph 5. (1) = If k is in A002182 itself, its index appears in this column. MN(k) = multiplicity notation of k; write the exponents of prime factors p_j of k in the j-th column, working left to right. Note: the order and population of each family k is not clear for i > 111, given the 10^3 terms at https://oeis.org/A002182/b002182.txt. An extended table was generated using 2^16 terms in [Flammenkamp 1] that confirms more than 1000 terms, but is too large to present here. k i pop. (1) MN(k) ---------------------------------------- 1 1 3 1 0 2 2 3 2 1 4 3 3 3 2 6 4 3 4 1.1 8 5 3 3 12 6 3 5 2.1 24 7 4 6 3.1 36 8 3 7 2.2 48 9 3 8 4.1 72 10 5 3.2 96 11 3 5.1 120 12 4 10 3.1.1 144 13 4 4.2 216 14 5 3.3 240 15 5 12 4.1.1 288 16 5 5.2 360 17 4 13 3.2.1 480 18 3 5.1.1 576 19 4 6.2 720 20 4 14 4.2.1 1080 21 4 3.3.1 1440 22 6 5.2.1 2160 23 3 4.3.1 2880 24 4 6.2.1 4320 25 5 5.3.1 5040 26 4 19 4.2.1.1 7200 27 3 5.2.2 7560 28 4 20 3.3.1.1 8640 29 3 6.3.1 10080 30 7 21 5.2.1.1 14400 31 5 6.2.2 15120 32 5 22 4.3.1.1 20160 33 6 23 6.2.1.1 30240 34 9 5.3.1.1 40320 35 6 7.2.1.1 50400 36 5 27 5.2.2.1 60480 37 8 6.3.1.1 90720 38 6 5.4.1.1 100800 39 8 6.2.2.1 120960 40 8 7.3.1.1 151200 41 8 5.3.2.1 181440 42 6 6.4.1.1 241920 43 6 8.3.1.1 302400 44 8 6.3.2.1 362880 45 6 7.4.1.1 453600 46 5 5.4.2.1 604800 47 7 7.3.2.1 907200 48 8 6.4.2.1 1209600 49 9 8.3.2.1 1330560 50 5 7.3.1.1.1 1663200 51 5 5.3.2.1.1 1814400 52 7 7.4.2.1 2419200 53 6 9.3.2.1 2661120 54 5 8.3.1.1.1 3326400 55 6 6.3.2.1.1 3991680 56 5 7.4.1.1.1 4989600 57 6 5.4.2.1.1 6350400 58 5 6.4.2.2 6652800 59 6 7.3.2.1.1 9979200 60 9 6.4.2.1.1 13305600 61 9 8.3.2.1.1 17297280 62 6 50 7.3.1.1.1.1 19958400 63 9 7.4.2.1.1 26611200 64 9 9.3.2.1.1 29937600 65 6 6.5.2.1.1 33264000 66 6 7.3.3.1.1 39916800 67 7 8.4.2.1.1 51891840 68 8 7.4.1.1.1.1 59875200 69 7 7.5.2.1.1 69854400 70 7 6.4.2.2.1 79833600 71 7 9.4.2.1.1 86486400 72 9 7.3.2.1.1.1 119750400 73 5 8.5.2.1.1 129729600 74 10 6.4.2.1.1.1 172972800 75 10 8.3.2.1.1.1 239500800 76 5 9.5.2.1.1 259459200 77 13 7.4.2.1.1.1 345945600 78 9 9.3.2.1.1.1 389188800 79 9 6.5.2.1.1.1 432432000 80 8 7.3.3.1.1.1 518918400 81 10 8.4.2.1.1.1 778377600 82 10 7.5.2.1.1.1 908107200 83 7 6.4.2.2.1.1 1037836800 84 10 9.4.2.1.1.1 1210809600 85 8 8.3.2.2.1.1 1297296000 86 10 7.4.3.1.1.1 1556755200 87 10 8.5.2.1.1.1 1816214400 88 9 7.4.2.2.1.1 2421619200 89 8 9.3.2.2.1.1 2594592000 90 10 8.4.3.1.1.1 3113510400 91 10 9.5.2.1.1.1 3632428800 92 13 8.4.2.2.1.1 4410806400 93 10 7.4.2.1.1.1.1 5189184000 94 8 9.4.3.1.1.1 5448643200 95 9 7.5.2.2.1.1 7264857600 96 13 9.4.2.2.1.1 8821612800 97 7 8.4.2.1.1.1.1 10897286400 98 14 8.5.2.2.1.1 13232419200 99 7 7.5.2.1.1.1.1 14529715200 100 3 10.4.2.2.1.1 15567552000 101 7 9.5.3.1.1.1 17643225600 102 7 9.4.2.1.1.1.1 18162144000 103 10 8.4.3.2.1.1 21794572800 104 14 9.5.2.2.1.1 26464838400 105 7 8.5.2.1.1.1.1 30875644800 106 7 7.4.2.2.1.1.1 36324288000 107 10 9.4.3.2.1.1 43589145600 108 4 10.5.2.2.1.1 44108064000 109 7 8.4.3.1.1.1.1 52929676800 110 7 9.5.2.1.1.1.1 54486432000 111 11 8.5.3.2.1.1 +-------+ | Code. | +-------+ Mathematica code appears below if you would like to explore these latter charts on your own. You must first download the b-file at A002182: visit https://oeis.org/A002182/b002182.txt. In this file on your local drive, delete the first non-data lines. (* Multiplicity notation function *) A054841[n_] := If[n == 1, {0}, Function[f, ReplacePart[Table[0, {PrimePi[f[[-1, 1]]]}], #] &@ Map[PrimePi@ First@ # -> Last@ # &, f]]@ FactorInteger@ n]; (* Grid-style chart similar to Table 4. *) Block[{s = Import["b002182.txt", "Data"][[All, -1]], a, b, c, m, u}, s = Take[s, 1000]; a = Array[{#2, #1, StringTrim[ StringReplace[ToString@ #, ", " -> "."], ("{" | "}") ...] &[#3 /. {} -> 0], Times @@ MapIndexed[Prime[First@ #2]^#1 &, #3]} & @@ {#1, Boole[First@ #2 > 0] Length@ #2, DeleteCases[-1 + #2, 0] /. -1 -> 0} & @@ {s[[#]], A054841@ s[[#]]} &, Length@ s]; u = Function[{t, k}, Map[StringPadRight[#, k] &, t]] @@ {#, Max@ Map[StringLength, #]} &@ Map[StringTrim[ StringReplace[ToString@ A054841@ #, ", " -> "."], ("{" | "}") ...] &, Union@ a[[All, -1]] ]; b = MapIndexed[{i_, j_, k_, #1} -> ToExpression@ StringJoin["{i,", ToString@ First@ #2, ",", " j, k}"] &, Union@ a[[All, -1]]]; c = Map[# /. b &, a]; m = Max[c[[All, 2]] ]; c = Map[Sort@ # &, SplitBy[SortBy[c, First], First]]; Grid[Transpose@ Prepend[Array[ With[{t = ConstantArray[0, m]}, {# - 1}~Join~ Map[If[# == 0, "", Item[If[IntegerQ@ #, Style[#, FontColor -> White], ""], Frame -> True, Background -> Red]] &, ReplacePart[t, Map[#2 -> If[#1 < 4, #3, "x"] & @@ # &, c[[#]] ]] ] ] &, Length@ c], Prepend[u, ""]], Frame -> All, FrameStyle -> Directive[LightGray, 12] ] ] (* Table 5 (without SHCN display) *) Block[{s = Import["b002182.txt", "Data"][[All, -1]], a, b, c, m, u}, s = Take[s, 1000]; a = Array[{#2, #1, StringTrim[ StringReplace[ToString@ #, ", " -> "."], ("{" | "}") ...] &[#3 /. {} -> 0], Times @@ MapIndexed[Prime[First@ #2]^#1 &, #3]} & @@ {#1, Boole[First@ #2 > 0] Length@ #2, DeleteCases[-1 + #2, 0] /. -1 -> 0} & @@ {s[[#]], A054841@ s[[#]]} &, Length@ s]; u = Function[{t, k}, Map[StringPadRight[#, k] &, t]] @@ {#, Max@ Map[StringLength, #]} &@ Map[StringTrim[ StringReplace[ToString@ A054841@ #, ", " -> "."], ("{" | "}") ...] &, Union@ a[[All, -1]] ]; b = MapIndexed[{i_, j_, k_, #1} -> ToExpression@ StringJoin["{i,", ToString@ First@ #2, ",", " j, k}"] &, Union@ a[[All, -1]]]; c = Map[# /. b &, a]; m = Max[c[[All, 2]] ]; c = Map[Sort@ # &, SplitBy[SortBy[c, First], First]]; MapIndexed[u[[First@ #2]]~StringJoin~StringJoin[#1] &, Transpose@ Array[With[{t = ConstantArray[" ", m]}, ReplacePart[t, Map[#2 -> "x" & @@ # &, c[[#]] ]] ] &, Length@ c]] ] // TableForm (* Table 6 *) Block[{s = Import["b002182.txt", "Data"][[All, -1]], a, b, c, m, u}, s = Take[s, 1000]; a = Array[{#2, #1, StringTrim[ StringReplace[ToString@ #, ", " -> "."], ("{" | "}") ...] &[#3 /. {} -> 0], Times @@ MapIndexed[Prime[First@ #2]^#1 &, #3]} & @@ {#1, Boole[First@ #2 > 0] Length@ #2, DeleteCases[-1 + #2, 0] /. -1 -> 0} & @@ {s[[#]], A054841@ s[[#]]} &, Length@ s]; u = Union@ a[[All, -1]] ; b = MapIndexed[{i_, j_, k_, #1} -> ToExpression@ StringJoin["{i,", ToString@ First@ #2, ",", " j, k}"] &, Union@ a[[All, -1]]]; c = Map[# /. b &, a]; m = Max[c[[All, 2]] ]; c = Map[Sort@ # &, SplitBy[SortBy[c, First], First]]; MapIndexed[{u[[#1]], If[#1 > 111, StringJoin["(", ToString@ #, ")"], #] &@ #1, If[#1 > 111, StringJoin["(", ToString@ #2, ")"], #2], If[FreeQ[s, u[[#1]] ], "", FirstPosition[s, u[[#1]]][[1]] ], StringTrim[ StringReplace[ToString@ A054841@ u[[#1]], ", " -> "."], ("{" | "}") ... ]} & @@ {First@ #2, Total@ #1} &, Transpose@ Array[With[{t = ConstantArray[0, m]}, ReplacePart[t, Map[#2 -> If[#1 < 4, 1, 1] & @@ # &, c[[#]] ]] ] &, Length@ c]] ] // TableForm Contact me on LinkedIn if you desire other programs I've used to generate this file for your own research. ----- Mathematica code to translate Achim Flammenkamp's data (renamed "HCN_124260.txt") to decimal: Times @@ {Times @@ Prime@ Range@ ToExpression@ First@ #1, If[# == {}, 1, Times @@ MapIndexed[Prime[First@ #2]^#1 &, #]] &@ DeleteCases[-1 + Flatten@ Map[ If[StringFreeQ[#, "^"], ToExpression@ #, ConstantArray[#1, #2] & @@ ToExpression@ StringSplit[#, "^"]] &, #2], 0]} & @@ TakeDrop[Drop[StringSplit@ #, 2], 1] & /@ Import["HCN_124260.txt", "Data"] ----- ////// Dedication ////// This work is dedicated to my father, Thomas De Vlieger, who had supported my computer science interests first with that 64k RAM Apple II+ in December 1980. It is by having this machine that I learned Applesoft BASIC on my own at ages 10-12 and taught myself how to program. It would not have happened were it not for his foresight, as I wanted to be an astronomer, and ended up majoring in architecture. Obviously his intuition demonstrated a level of knowledge of my true interests that proved even more honest and true than my own. Thank you Dad, and I love you. ----- ////// Acknowledgements ////// Many thanks are owed to Achim Flammenkamp for my long-standing interest in the data he'd generated regarding the highly composite numbers, especially my use of OEIS A054841 to abbreviate highly composite and highly regular numbers. Thanks are also in order for Dr. Neil Sloane and the OEIS for its ready availability of the mindshare of mathematical knowledge made available freely to the world. I had been interested in sequences since age 17 via references in the CRC Standard Math Tables, 28th Edition. It is through the OEIS that I attained fluency in Mathematica, and none of this work would thus be possible. ----- ////// References ////// [Flammenkamp 1]: Achim Flammenkamp, "Highly Composite Numbers", http://wwwhomes.uni-bielefeld.de/achim/highly.html retrieved 201803292330 GMT. Downloads appear at the bottom of page and require "gunzip" application. [Sloane 1]: Neil Sloane, The Online Encyclopedia of Integer Sequences, https://oeis.org, retrieved 20180328 (see "Concerns Sequences" at top for specific pages accesssed.) ----- ////// Revision Record ////// 201803281015 Created. 201803281330 A301892 extended to 115 terms. 201803281945 A301892 extended to 125 terms. 201803282030 Table 3 added. 201803282200 Table 4 added. 201803290830 A301892 extended to 135 terms. 201803290845 Converted Table 1 last column to MN( A002182(n)/A002110(A108602(n)) ). 201803291345 Graph 5 added. 201803291445 Table 6 added. 201803291500 Code added. 201803291700 Extensions to tables and codes. eof