s20210622 | Cycle Signature Parameters for Phase (1a). Michael De Vlieger, St. Louis, Missouri, 202106271530. This data pertains to terms in the integer sequence s20210622. (See paper http://www.vincico.com/seq/s20210622.html.) A cycle c(i) in sequence a = s20210622 begins at a(k) with the fruit of Condition 0 (i.e., novel m -> tau(m)), and the subsequent run of Condition 1 (i.e., extant m => (m + tau(u))). The "triple point" = (n, a(n)) = (414, 14478), in cycle c(385). The "Signature" column writes the signature (a(k), a(k+1)-a(k)) written as "a(k)+(a(k+1)-a(k))". For instance, the signature "4+32" has the first terms a(k) = 4, a(k+1) = 36. An asterisk signifies a signature that appears in the "gorges". Qty. refers to the number of cycles with the signature for 1 <= n <= 2^20. i is the index of the first cycle c(i) that has the signature. j is the index of the last known cycle c(j) that has the signature. p shows an asterisk "*" if the signature is extinct before i = 385, or a full stop "." if the signature was seen first at i < 385. k is the index in c(i) of the term 414, or - if 414 doesn't appear in the sequence. s is the number of perfect squares m in the cycle trajectory for m <= 414. sq. sig. is the square signature of the cycle-signature. It is the decimal equivalent of the binary compactification of the square roots of all square m. The roots of squares present is a diagrammatic depiction where the n-th column bears an asterisk if n is squared in the cycle signature. Signature Qty. i j p k s sq. sig. Roots of squares present ------------------------------------------------------------------------------- 1+1 1 1 1 * 69 10 25853 *.******..*..**..... 2+1 1 2 2 * 68 9 25852 ..******..*..**..... 2+2 18 3 2235 . 65 4 78 .***..*............. 2+3 3 8 68 * 67 9 25852 ..******..*..**..... 2+4 19 36 2942 . 76 2 4104 ...*........*....... 2+5 1 27 27 * 66 9 25852 ..******..*..**..... 2+6 9 32 2968 . 62 4 25728 .......*..*..**..... 2+8 13 40 2963 . 75 2 4104 ...*........*....... 2+9 5 236 2818 64 1 64 ......*............. 2+10 2 114 149 * 54 2 1536 .........**......... 2+11 2 700 703 . 63 1 64 ......*............. 2+12 10 100 2969 . 63 8 25848 ...*****..*..**..... 2+15 5 329 3041 62 1 64 ......*............. 2+16 2 212 2724 . 66 2 4128 .....*......*....... 2+20 1 1384 1384 . 65 2 4128 .....*......*....... 2+21 1 1753 1753 . 56 1 16 ....*............... 2+25 1 847 847 . 58 1 64 ......*............. 2+27 5 622 3162 58 1 64 ......*............. 3+2 3 4 47 * 67 9 25852 ..******..*..**..... 3+5 1 28 28 * 62 4 25728 .......*..*..**..... 3+6 1 131 131 * 65 9 25852 ..******..*..**..... 3+12 1 1882 1882 . 62 1 64 ......*............. 3+15 1 327 327 * 66 2 4128 .....*......*....... 3+16 1 158 158 * 61 1 64 ......*............. 4+2 19 6 2943 . 64 4 78 .***..*............. 4+3 5 7 213 * 76 3 4106 .*.*........*....... 4+4 97 15 3146 64 2 66 .*....*............. 4+6 17 29 2118 . 64 2 66 .*....*............. 4+7 1 90 90 * 64 9 25850 .*.*****..*..**..... 4+8 140 93 3234 59 4 402 .*..*..**........... 4+9 5 235 2817 62 3 74 .*.*..*............. 4+10 5 67 2707 . 62 2 66 .*....*............. 4+12 80 101 2867 57 2 10 .*.*................ 4+14 2 577 928 . 73 2 4098 .*..........*....... 4+15 4 549 3039 66 3 4130 .*...*......*....... 4+16 100 252 3242 60 2 66 .*....*............. 4+18 9 223 2945 . 57 4 402 .*..*..**........... 4+20 15 273 3131 57 2 66 .*....*............. 4+21 1 1751 1751 . 71 3 4114 .*..*.......*....... 4+24 43 388 3195 70 2 4098 .*..........*....... 4+25 2 845 846 . 55 3 386 .*.....**........... 4+27 5 623 3163 59 7 25826 .*...***..*..**..... 4+28 3 1075 2855 68 2 4098 .*..........*....... 4+32 * 25 583 3224 - 4 264226 .*...*.....*......*. 4+36 6 1874 3076 64 2 4098 .*..........*....... 4+48 2 2420 2808 62 2 4098 .*..........*....... 5+4 2 11 80 * 65 9 25852 ..******..*..**..... 6+2 6 10 178 * 75 2 4104 ...*........*....... 6+4 44 18 2986 61 4 25728 .......*..*..**..... 6+5 1 115 115 * 63 1 64 ......*............. 6+6 28 21 3010 74 2 4104 ...*........*....... 6+8 31 75 2974 . 60 4 25728 .......*..*..**..... 6+9 5 234 2069 58 1 16 ....*............... 6+10 9 66 2778 62 8 25848 ...*****..*..**..... 6+11 1 702 702 . 61 1 64 ......*............. 6+12 29 83 2903 59 4 25728 .......*..*..**..... 6+14 7 323 3000 52 2 1536 .........**......... 6+16 30 296 2979 55 . 0 .................... 6+18 11 281 3043 64 2 4128 .....*......*....... 6+20 5 274 2635 54 . 0 .................... 6+21 1 1752 1752 . 71 1 4096 ............*....... 6+24 13 530 2729 . 53 . 0 .................... 6+30 1 788 788 . 57 6 25824 .....***..*..**..... 6+32 3 2580 2907 52 . 0 .................... 6+36 1 2528 2528 51 . 0 .................... 7+2 1 58 58 * 65 9 25852 ..******..*..**..... 8+2 10 23 2962 . 61 4 25728 .......*..*..**..... 8+3 1 353 353 * 63 1 64 ......*............. 8+4 92 41 3241 74 2 4104 ...*........*....... 8+6 27 34 3148 60 4 25728 .......*..*..**..... 8+7 1 89 89 * 58 1 16 ....*............... 8+8 * 216 74 3243 62 8 25848 ...*****..*..**..... 8+9 1 2816 2816 61 1 64 ......*............. 8+10 13 113 2482 59 4 25728 .......*..*..**..... 8+12 * 115 102 3247 52 2 1536 .........**......... 8+14 5 232 2386 55 . 0 .................... 8+15 5 328 1491 . 59 1 64 ......*............. 8+16 172 211 3229 64 2 4128 .....*......*....... 8+18 8 222 2598 54 . 0 .................... 8+20 18 631 3145 57 4 25728 .......*..*..**..... 8+24 80 352 3194 63 2 4128 .....*......*....... 8+27 1 624 624 . 55 1 64 ......*............. 8+28 6 662 3064 57 6 25824 .....***..*..**..... 8+30 4 690 2745 52 . 0 .................... 8+32 35 590 3206 52 2 384 .......**........... 8+36 9 1115 2768 54 4 25728 .......*..*..**..... 8+40 * 4 2936 3009 - 2 264192 ...........*......*. 8+48 3 2782 3088 46 2 1536 .........**......... 9+4 1 35 35 * 62 3 76 ..**..*............. 9+6 3 383 1131 . 60 5 25732 ..*....*..*..**..... 9+8 3 197 853 . 60 5 25732 ..*....*..*..**..... 9+10 1 248 248 * 66 3 4132 ..*..*......*....... 9+12 2 128 1142 . 65 3 4132 ..*..*......*....... 9+16 1 2653 2653 71 3 4116 ..*.*.......*....... 9+20 1 2029 2029 55 3 388 ..*....**........... 10+4 12 50 2387 60 4 25728 .......*..*..**..... 10+6 7 253 2277 62 8 25848 ...*****..*..**..... 10+8 20 506 3201 59 4 25728 .......*..*..**..... 10+11 1 701 701 . 57 3 400 ....*..**........... 10+12 7 144 2660 55 . 0 .................... 10+16 7 817 2934 54 . 0 .................... 10+18 6 282 3092 57 4 25728 .......*..*..**..... 10+20 1 1582 1582 . 53 . 0 .................... 10+24 2 351 588 . 53 . 0 .................... 10+32 1 1432 1432 . 51 . 0 .................... 10+48 1 2679 2679 49 . 0 .................... 11+16 1 599 599 . 58 1 64 ......*............. 12+2 2 1375 2334 . 53 2 1536 .........**......... 12+3 2 69 159 * 58 3 400 ....*..**........... 12+4 58 123 3068 73 2 4104 ...*........*....... 12+6 51 105 3058 52 2 1536 .........**......... 12+7 1 91 91 * 56 1 16 ....*............... 12+8 114 78 3244 65 2 4128 .....*......*....... 12+9 1 54 54 * 72 1 4096 ............*....... 12+10 20 773 3120 58 4 25728 .......*..*..**..... 12+12 * 56 84 3158 51 2 1536 .........**......... 12+14 3 233 2441 64 2 4128 .....*......*....... 12+15 2 2120 2422 70 1 4096 ............*....... 12+16 63 256 3230 54 2 384 .......**........... 12+18 20 400 3149 64 2 4128 .....*......*....... 12+20 7 2018 2900 56 4 25728 .......*..*..**..... 12+21 1 1000 1000 . 55 1 64 ......*............. 12+22 3 1710 1712 . 53 2 384 .......**........... 12+24 33 529 3113 62 2 4128 .....*......*....... 12+28 2 660 661 . 51 . 0 .................... 12+30 3 689 2157 51 . 0 .................... 12+32 8 862 3013 52 . 0 .................... 12+36 3 1760 2981 53 4 25728 .......*..*..**..... 14+2 1 347 347 * 62 8 25848 ...*****..*..**..... 14+4 2 863 1900 . 59 4 25728 .......*..*..**..... 14+6 4 196 1988 52 2 1536 .........**......... 14+8 3 1053 2612 55 . 0 .................... 14+12 2 1177 2309 54 . 0 .................... 14+14 1 1528 1528 . 57 4 25728 .......*..*..**..... 14+16 4 524 2327 53 . 0 .................... 14+18 2 713 1472 . 63 2 4128 .....*......*....... 14+24 2 799 2077 52 . 0 .................... 14+30 2 2744 3199 54 4 25728 .......*..*..**..... 15+2 1 166 166 * 61 1 64 ......*............. 15+8 4 279 2894 59 1 64 ......*............. 15+12 1 1147 1147 . 71 1 4096 ............*....... 15+16 1 2325 2325 70 1 4096 ............*....... 15+18 1 1454 1454 . 69 1 4096 ............*....... 15+32 1 1438 1438 . 63 1 4096 ............*....... 16+4 42 173 3239 61 8 25848 ...*****..*..**..... 16+6 28 130 3106 72 2 4104 ...*........*....... 16+8 107 227 3233 70 2 4104 ...*........*....... 16+10 4 205 3025 71 2 4104 ...*........*....... 16+12 48 242 3217 56 2 72 ...*..*............. 16+14 2 735 1581 . 55 2 72 ...*..*............. 16+16 68 439 3240 68 2 4104 ...*........*....... 16+18 13 313 3211 55 2 72 ...*..*............. 16+20 11 611 2807 53 4 424 ...*.*.**........... 16+22 1 1713 1713 . 54 2 72 ...*..*............. 16+24 43 603 3237 63 2 4104 ...*........*....... 16+28 4 874 2800 52 2 72 ...*..*............. 16+30 10 1340 3200 62 2 4104 ...*........*....... 16+32 15 1170 3223 60 2 4104 ...*........*....... 16+36 6 1405 2687 60 2 4104 ...*........*....... 16+40 1 2959 2959 59 2 4104 ...*........*....... 18+2 2 221 902 . 65 2 4128 .....*......*....... 18+4 9 283 2989 58 4 25728 .......*..*..**..... 18+6 7 338 2278 51 2 1536 .........**......... 18+8 15 176 2722 . 64 2 4128 .....*......*....... 18+10 2 357 1315 . 54 2 384 .......**........... 18+12 5 515 2352 . 64 2 4128 .....*......*....... 18+14 4 1580 2442 56 4 25728 .......*..*..**..... 18+16 13 334 3048 53 2 384 .......**........... 18+18 8 575 3151 62 2 4128 .....*......*....... 18+20 5 569 3130 55 4 25728 .......*..*..**..... 18+24 2 1258 1635 . 51 . 0 .................... 18+30 2 789 1158 . 53 4 25728 .......*..*..**..... 18+32 1 1929 1929 . 51 . 0 .................... 20+4 7 343 2121 . 51 2 1536 .........**......... 20+6 5 294 3069 64 2 4128 .....*......*....... 20+8 17 208 2982 54 2 384 .......**........... 20+10 7 427 2917 64 2 4128 .....*......*....... 20+12 9 441 3157 56 4 25728 .......*..*..**..... 20+14 1 1151 1151 . 53 2 384 .......**........... 20+16 11 600 3212 62 2 4128 .....*......*....... 20+18 7 868 2599 55 4 25728 .......*..*..**..... 20+20 1 2992 2992 51 . 0 .................... 20+24 6 968 2125 52 . 0 .................... 20+30 2 994 2164 51 . 0 .................... 20+32 1 3167 3167 50 2 384 .......**........... 20+36 1 1116 1116 . 50 . 0 .................... 20+48 3 2680 2846 47 . 0 .................... 21+6 1 1135 1135 . 71 1 4096 ............*....... 21+8 2 1603 1696 . 57 1 64 ......*............. 21+12 1 932 932 . 69 1 4096 ............*....... 22+18 1 1680 1680 . 52 2 384 .......**........... 24+4 25 375 3190 63 2 4128 .....*......*....... 24+6 18 499 3132 50 2 1536 .........**......... 24+8 51 380 3210 52 . 0 .................... 24+9 1 2068 2068 67 1 4096 ............*....... 24+10 6 827 3156 52 . 0 .................... 24+12 24 340 3073 63 2 4128 .....*......*....... 24+14 2 1862 2700 49 2 1536 .........**......... 24+16 34 291 3001 51 . 0 .................... 24+18 7 712 2604 51 2 384 .......**........... 24+20 5 630 2899 51 2 384 .......**........... 24+24 11 1331 3182 50 . 0 .................... 24+27 1 3164 3164 60 1 4096 ............*....... 24+28 1 3115 3115 48 . 0 .................... 24+30 6 1413 3189 50 . 0 .................... 24+32 12 1357 3205 52 4 25728 .......*..*..**..... 24+36 3 823 2882 45 2 1536 .........**......... 24+40 4 1739 3008 51 4 25728 .......*..*..**..... 25+24 1 774 774 . 63 3 4176 ....*.*.....*....... 27+4 1 3056 3056 70 1 4096 ............*....... 27+8 1 1296 1296 . 55 1 64 ......*............. 27+12 1 1043 1043 . 54 1 64 ......*............. 27+16 1 525 525 . 53 1 64 ......*............. 28+4 3 1282 2881 56 4 25728 .......*..*..**..... 28+6 1 1033 1033 . 53 2 384 .......**........... 28+8 6 1847 2993 62 2 4128 .....*......*....... 28+10 1 812 812 . 55 4 25728 .......*..*..**..... 28+16 2 1732 2620 52 . 0 .................... 28+18 1 1470 1470 . 52 . 0 .................... 28+20 1 566 566 . 53 4 25728 .......*..*..**..... 30+4 5 1292 2357 52 . 0 .................... 30+6 4 721 2076 63 2 4128 .....*......*....... 30+8 4 944 2898 49 2 1536 .........**......... 30+12 2 596 1410 . 51 2 384 .......**........... 30+16 1 1055 1055 . 50 . 0 .................... 30+18 1 3150 3150 50 . 0 .................... 30+20 1 2976 2976 50 2 384 .......**........... 30+24 2 467 1973 50 . 0 .................... 30+30 1 3055 3055 45 2 1536 .........**......... 30+32 1 2655 2655 49 . 0 .................... 30+40 1 2960 2960 46 . 0 .................... 32+4 8 1019 3225 62 2 4128 .....*......*....... 32+6 3 801 1830 . 55 4 25728 .......*..*..**..... 32+8 13 1531 3129 51 . 0 .................... 32+10 3 866 2449 51 . 0 .................... 32+12 4 516 2879 52 . 0 .................... 32+14 1 930 930 . 52 . 0 .................... 32+16 11 1220 3121 53 4 25728 .......*..*..**..... 32+18 3 2601 3152 51 . 0 .................... 32+20 2 610 2030 50 2 384 .......**........... 32+21 1 1746 1746 . 61 1 4096 ............*....... 32+24 3 1063 1833 . 50 . 0 .................... 32+30 * 2 2146 2158 . - 2 264192 ...........*......*. 32+32 3 1436 2905 55 2 4224 .......*....*....... 32+36 1 1449 1449 . 47 . 0 .................... 32+40 * 1 1740 1740 . - 2 264192 ...........*......*. 36+4 7 759 2789 56 6 25824 .....***..*..**..... 36+6 3 1829 2939 61 2 4128 .....*......*....... 36+8 9 857 2916 62 2 4128 .....*......*....... 36+10 2 2364 2689 61 2 4128 .....*......*....... 36+12 1 2394 2394 60 2 4128 .....*......*....... 36+16 4 1307 2631 60 2 4128 .....*......*....... 36+18 1 2603 2603 58 2 4128 .....*......*....... 36+22 1 1714 1714 . 59 2 4128 .....*......*....... 36+24 2 1312 1835 . 48 2 288 .....*..*........... 36+30 2 1010 2165 52 4 25632 .....*....*..**..... 36+32 1 2694 2694 55 2 4128 .....*......*....... 40+4 1 2287 2287 51 2 384 .......**........... 40+8 3 2697 3179 50 . 0 .................... 40+10 1 1456 1456 . 50 2 384 .......**........... 40+12 2 1689 2317 48 . 0 .................... 40+14 1 2701 2701 50 . 0 .................... 40+16 3 1707 2864 52 4 25728 .......*..*..**..... 45+8 1 1916 1916 61 1 4096 ............*....... 48+4 * 1 2931 2931 . - 2 264192 ...........*......*. 48+6 1 3123 3123 52 4 25728 .......*..*..**..... 48+8 5 2365 2990 49 . 0 .................... 48+12 1 2004 2004 47 . 0 .................... 48+16 2 2652 2737 48 2 384 .......**........... 48+18 1 1455 1455 . 45 . 0 .................... 48+24 2 2304 2588 45 . 0 .................... 48+30 1 3099 3099 - 2 264192 ...........*......*. 50+8 1 3160 3160 49 2 384 .......**........... 54+18 1 3153 3153 44 . 0 .................... 60+6 1 2413 2413 46 . 0 ....................