Vinci Constitutive Number Sequence Catalog (V-series sequences). Since 18 May 2018. Updated 20 September 2024 Created to promote a convenient framework for constitutive mathematics, not to supplant OEIS. A-numbers appear when V-numbers are in OEIS. A bullet indicates a sequence that to my knowledge isn't in OEIS and hence represents an original sequence. (Merely that a sequence is "original" doesn't mean that someone else hasn't thought about such a sequence.) The Vinci Constitutive Catalog is an aid created and maintained by Michael Thomas De Vlieger. We reserve the right to reposition IDs that are less than a year old to allow the catalog to function efficiently. A001477: V0000: The nonnegative integers. A000040: V0001: The primes, { p : Ω(p) = ω(p) = 1 }. A001597: V0002: Perfect powers, { k^δ, δ > 1 }. A001694: V0003: Powerful numbers, { a² × b³ : a ≥ 1 ∧ b ≥ 1 }. A246547: V0004: Multus numbers, i.e., composite prime powers, { k : Ω(n) > ω(n) = 1 }. A005117: V0005: Squarefree numbers, { k : Ω(n) = ω(n) ≥ 1 }. A120944: V0006: Varius numbers, i.e., composite squarefree numbers, { k : Ω(n) = ω(n) > 1 }. A126706: V0007: Tantus numbers, i.e., { V3 \ V4 } = { k : Ω(n) > ω(n) > 1 }. A286708: V0008: Plenus numbers, i.e., products of multus numbers, V8 ⊂ V7, { k : rad(k)² | k, ω(k) > 1 }. A000961: V0009: Prime powers (including the empty product), { k : Ω(n) = ω(n) ≥ 0 }. A027750: V0010: list of divisors of n. A000005: V0011: divisor counting function τ(n). A002182: V0012: indices of records in τ(n) (highly composite numbers). A002183: V0013: records in τ(n). A002201: V0014: superior highly composite numbers. A173540: V0015: list of nondivisors of n. A049820: V0016: nondivisor counting function. A000040: V0017: indices of records in A049820 (the primes). A040976: V0018: records in A049820 (prime p − 2). A051731: V0019: characteristic function of divisors of n. A038566: V0020: list of totatives of n (reduced residue system of n). A000010: V0021: Euler totient function φ(n). A006005: V0022: indices of records in A000010 (the odd primes together with 1). A006093: V0023: records in A000010 (prime p – 1). A121998: V0025: list of cototatives of n. A051953: V0026: cototient function. A063985: V0027: indices of records in A051953. A063986: V0028: records in A051953. A054521: V0029: characteristic function of totatives of n. A162306: V0030: list of 1 ≤ k ≤ n such that k | ne with e ≥ 0. A010846: V0031: regular counting function θ(n). A244052: • V0032: indices of records in A010846. “Highly regular numbers.” A244053: • V0033: records in A010846. A008479: V0034: coregular counting function, | { k : k ≤ n, rad(k) = rad(n) } | A279907: • V0035: richness e of 1 ≤ k ≤ n. A280269: • V0036: richness e of regulars 1 ≤ k ≤ n. A280274: • V0037: maximum richness e of regulars 1 ≤ k ≤ n. A294306: • V0038: population of richnesses e of regulars 1 ≤ k ≤ n. A304569: • V0039: characteristic function of regulars of n. A133995: V0040: list of neutrals k < n. A045763: V0041: neutral counting function (ncf(n), ξ(n)). A300859: • V0042: indices of records in A045763. A300914: • V0043: records in A045763. A294492: • V0044: indices of records for ratio A045763(n)/n. A295523: • V0045: nonprime numbers with A243822(n) ≥ A243823(n). (finite, 7 terms). A300858: • V0046: A243823(n) − A243822(n), i.e., ξt (n) − ξd (n). — : • V0047: — : • V0048: A304571: • V0049: characteristic function of neutrals of n. A272618: • V0050: list of 1 ≤ k ≤ n such that k | ne with e > 1. A243822: • V0051: semidivisor counting function ξd (n). A293555: • V0052: indices of records in A243822. “Highly semidivisible numbers.” A293556: • V0053: records in A243822. A299990: • V0054: A243822(n) − A000005(n) = ξd (n) − τ(n). A289280: V0055: Smallest k > n regular to n. A355432: • V0056: Symmetric Semidivisor Counting Function ξ₉(n). (->V5101) A360589: • V0057: Indices of records in ξ₉(n) V5102 ⊂ V0210. (->V5102) A360912: • V0058: Records in ξ₉(n). (->V5103) A304570: • V0059: characteristic function of semidivisors of n. A272619: • V0060: list of semitotatives. A243823: • V0061: semitotative counting function ξt (n). A292867: • V0062: indices of records in A243823. “Highly semitotative numbers.” A292868: • V0063: records in A243823. A294575: • V0064: semitotative-dominant numbers (2 ×ξt (n) > n). A291989: • V0065: Smallest k > n semicoprime to n. A295221: • V0067: Semitotative parity numbers (2 ×ξt (n) = n). A096014: • V0068: smallest number m = pq semicoprime to n, i.e., prime p | n, and q the smallest prime nondivisor of n, with a(1) = 2. A304572: • V0069: characteristic function of semitotatives of n. A366825: • V0070: Minimally tantus numbers { k : k = p²m : Ω(m) = ω(m) > 1, m > 1 } = V75 U V0771. (->V0701) A366250: • V0071: Carens-Liquidus ∪ { k = m × rad(k) : ω(m) ≠ ω(k) > 1, rad(m) | rad(k), k ≥ rad(k)² }. A303946: V0072: Non-valens A126706 \ A131605 = V7 \ V84. A364702: • V0073: Carens A361098 \ A286708 = V74 \ V8. (->V0740->V0749). A361098: • V0074: Panstitutive numbers, thick-strong tantus, A360765 ∩ A360768 = V0700 ∩ V0702. A364999: • V0075: Even minimally tantus numbers, thin-weak tantus, A363082 ∩ A360767 = V0701 ∩ V0703 = V70 \ V0771. A364998: • V0076: Thin-strong tantus, A363082 ∩ A360768 = V0700 ∩ V0703. A364997: • V0077: Thick-weak tantus, A360765 ∩ A360767 = V0701 ∩ V0702 = V0771 ∪ V0770. A364996: • V0078: Nonpanstitutive tantus, A363082 ∪ A360767 = A126708 \ A361098 = V7 \ V74. A332785: V0079: Carens numbers V7 \ V8. A303606: V0080: Planus numbers, varius numbers at multiplicity: { v^δ : v ∈ A120944, δ > 1 }. A131605: V0081: Perfect-power plenus (valens) numbers, A286708 ∩ A1597 = V8 ∩ V2 = V81 ∪ V86. (->V84) A366854: • V0082: Fortis numbers, tantus numbers at multiplicity: { t^δ : t ∈ A126706, δ > 1 } = A131605 \ A303606 = V81 \ V80. (->V83) A365308: • V0083: Perfect powers of composite primorials, primorials at multiplicity, A100778 ∩ A303606. (->V81) A177492: • V0084: { ϰ² : Ω(ϰ) = ω(ϰ) > 1 }. Squares of varius numbers. — : • V0085: A286708 \ A177492 = V8 \ V84 = {k ∈ V8 : k/rad(k) is not squarefree } A370266: • V0086: Union of A366250 and A286708 = k ∈ A024619 : k/rad(k) > rad(k). A365745: • V0087: A303606 \ A365308 = V80 \ V81. (->V0839->V0809) A052486: • V0088: Achilles numbers, A286708 \ A1597 = V8 \ V81 = V8 \ V2. (->V85) A359280: • V0089: Citroque numbers: V8 \ V80 = A286708 \ A303606. A007395: V0100: {2} = V1 ∩ V12 = A40 ∩ A2182. A002808: V0101: Composites. A013929: V0102: Nonsquarefree numbers. A006881: V0103: Squarefree semiprimes. A085971: * V0104 Nonmultus numbers. ℕ \ A246547. * (A085971 = ℕ \ {{1} ∪ A246547}.) A013939: V0105 Nonsquarefree numbers. ℕ \ A5117. A363597: V0106 Nonvarius numbers. ℕ \ A120944. A303554: V0107 Nontantus numbers. ℕ \ A126706. — : • V0108 Nonplenus numbers. ℕ \ A286708. A024619: V0109 Non-prime powers. ℕ \ A961. — : • V0110: {2, 3}, Primes not ±1 (mod 6). A002110: V0111: The primorials, products pn# of the smallest n primes. A006939: V0112: Superprimorials. A368507: V0113: Powers of superprimorials. A060735: V0114: k × pn# with 1 ≤ k < p(n + 1). A100778: V0115: Powers of primorials. A322793: V0116: Perfect powers of primorials. A347284: • V0117: Idaho numbers: a(n) = Product_{j=1..ℓ} pjej with 0 < ej < ⌊log(p(j−1)/log(pj))⌋. A168263: V0118: A2182 ∩ A060735 = V12 ∩ V0114 = {1, 2, 4, 6, 12, 24, 60, 120, 180, 840, 1260, 1680, 27720}. A053669: V0119: Smallest prime q that is coprime to n. A001248: V0120: Prime Squares. A007510: V0121: Single or non-twin primes. A001359: V0122: Lesser of twin primes. A006512: V0123: Greater of twin primes. A002144: V0124: Pythagorean primes, i.e., primes p mod 4 ≡ 1 A002145: V0125: Primes p mod 4 ≡ -1. A002476: V0126: Primes p mod 6 ≡ 1. A007528: V0127: Primes p mod 6 ≡ -1. A367511: • V0128: A2182 ∩ A341745 = V12 ∩ V3301 = {1, 4, 36, 48, 45360, 50400}. "Karl's Trace" numbers. A079067: V0129: Number of primes q < gpf(n) that do not divide n. A000720: V0131: Number of primes ≤ n, ⊠V0161, π(n). A069623: V0132: Number of perfect powers ≤ n, ⊠V0162, card({ k^δ, δ > 1, k^δ ≤ n }). A217038*: V0133: Number of powerful numbers ≤ n, ⊠V0163, card({ k = a² × b³ : a ≥ 1 ∧ b ≥ 1, k ≤ n }). A069637: V0134: Number of multus numbers ≤ n, ⊠V0164, card({ k : Ω(n) > ω(n) = 1, k ≤ n }). A013928: V0135: Number of squarefree numbers ≤ n, ⊠V0165, card({ k : Ω(n) = ω(n) ≥ 1, k ≤ n }). — : • V0136: Number of varius numbers ≤ n, ⊠V0166. — : • V0137: Number of tantus numbers ≤ n, ⊠V0167. — : • V0138: Number of plenus numbers ≤ n, ⊠V0168. A065515: V0139: Number of prime powers (including the empty product) ≤ n, ⊠V0169. A007053: V0141: Number of primes ≤ 2^n, π(2^n). A188951: V0142: Number of perfect powers ≤ 2^n, card({ k^δ, δ > 1, k^δ ≤ 2^n }). A062762: V0143: Number of powerful numbers ≤ 2^n, card({ k = a² × b³ : a ≥ 1 ∧ b ≥ 1, k ≤ 2^n }). — : • V0144: Number of multus numbers ≤ 2^n, card({ k : Ω(n) > ω(n) = 1, k ≤ 2^n }). A143658: V0145: Number of squarefree numbers ≤ 2^n, card({ k : Ω(n) = ω(n) ≥ 1, k ≤ 2^n }). — : • V0146: Number of varius numbers ≤ 2^n. A372403: • V0147: Number of tantus numbers ≤ 2^n. — : • V0148: Number of plenus numbers ≤ 2^n. A182908: V0149: Number of prime powers (including the empty product) ≤ 2^n. A000849: V0151: Number of primes ≤ P(n), π(P(n)). — : • V0152: Number of perfect powers ≤ P(n), card({ k^δ, δ > 1, k^δ ≤ P(n) }). — : • V0153: Number of powerful numbers ≤ P(n), card({ k = a² × b³ : a ≥ 1 ∧ b ≥ 1, k ≤ P(n) }). — : • V0154: Number of multus numbers ≤ P(n), card({ k : Ω(n) > ω(n) = 1, k ≤ P(n) }). A158341: V0155: Number of squarefree numbers ≤ P(n), card({ k : Ω(n) = ω(n) ≥ 1, k ≤ P(n) }). — : • V0156: Number of varius numbers ≤ P(n). — : • V0157: Number of tantus numbers ≤ P(n). — : • V0158: Number of plenus numbers ≤ P(n). — : • V0159: Number of prime powers (including the empty product) ≤ P(n). A010051: V0161: Characteristic function of primes. A075802: V0162: Characteristic function of perfect powers. A112526: V0163: Characteristic function of powerful numbers. A268340: V0164: Characteristic function of multus numbers. A008966: V0165: Characteristic function of squarefree numbers. A354819*: V0166: Characteristic function of varius numbers. * Includes 1. A355447: V0167: Characteristic function of tantus numbers. — : • V0168: Characteristic function of plenus numbers. A010055: V0169: Characteristic function of of prime powers (including the empty product). A008407: V0170: Minimal difference s(n) between beginning and end of prime n-tuplets permitted by divisibility considerations. A083409: V0171: Number of prime k-tuplet constellations. A186634: V0172: Irregular triangle, read by rows, giving gaps that describe dense patterns of n primes. A186702: V0173: Irregular tri. by rows, of the smallest prime starting a prime constellation in row n of A186634. A067392: V0190: Cototient sum function. A000203: V0191: Divisor sum function σ(n). A002093: V0192: Indices of records in σ(n) (highly abundant numbers). A034885: V0193: Records in σ(n). A004394: V0194: Superabundant numbers. (->V0512) A004490: V0195: Colossally abundant numbers. (->V0514) A001157: V0196: Sum of squares of divisors. A193988: V0197: Indices of records in A001157. A001158: V0198: Sum of cubes of divisors. A193989: V0199: Indices of records in A001157. A027746: V0200: Prime factors of n with multiplicity. A027748: V0201: Distinct prime factors of n (->V0200). A001221: V0202: (little) ω(n) = number of distinct prime factors of n (->V0201). A001222: V0203: (big) Ω(n) = number of distinct prime factors of n counting multiplicity (->V0202). A055932: V0210: products of a contiguous set of the smallest primes, with multiplicity. A025487: V0211: products of primorials (least integer of each prime signature). A056808: V0212: A055932 \ A025487. (->V0213->V0731) A363280: • V0213: A025487 \ A2182 = V0211 \ V12. (->V0212) A333655: V0214: A2182 \ A2201 = V12 \ V14. A368325: • V0215: A79 U A2110 = V0111 U V0401. A368681: • V0216: A025487 ∩ A1597 = V0401 U V0842 \ {2} = V2 ∩ V211 \ {2}. — : V0217: A369361: • V0218: ℕ \ A025487 = A080259 U A056808 = V0212 U V0219. A080259: V0219: ℕ \ A055932. A007947: V0220: squarefree kernel of n: largest squarefree number k | n, rad(k). A124010: V0221: prime signature of n. A003557: V0222: kernel ratio, k/rad(k). A064549: V0223: kernel product, k × rad(k). A369690: • V0224: max(A119288(n), A053669(n)). A280363: • V0230: floor(logp(n)), with p the least prime that divides n. A349317: • V0240: Triangle T(n,k) = [gcd(n, k) > 1], Iverson brackets. A349297: • V0241: Quincunx sequence: T(n,k) = [(2 | n ∧ 2 | k) ∨ (3 | n ∧ 3 | k)]. — : • V0242: 5-Quincunx sequence: T(n,k) = [(2 | n ∧ 2 | k) ∨ (3 | n ∧ 3 | k) ∨ (5 | n ∧ 5 | k)]. — : • V0243: 7-Quincunx sequence: T(n,k) = [∨_{i=1..4} (prime(i) | n ∧ prime(i) | k)]. — : • V0244: 11-Quincunx sequence: T(n,k) = [∨_{i=1..5} (prime(i) | n ∧ prime(i) | k)]. A349298: • V0245: Positions k in row n of triangles S(n,k) = T(n,k) = 0, where A054521 = S and A349297 = T, or 0 if there are no such k. — : • V0246: Positions k in row n of triangles S(n,k) = T(n,k) = 0, where A054521 = S and V0242 = T, or 0 if there are no such k. A375975: V0300: Lean numbers: Minimal θ(m) for m, product of k consecutive distinct prime factors. {1, 2, 5, 15, 50, 176, 641, (2374, < 8943, < 33818)} A052485: V0301: “weak” numbers, numbers that are not powerful, ℕ \ V3 = ℕ \ A1694. (->V0309) A357910: V0302: T(n,k), lex order natural numbers, factors written in decreasing order. k is coregular sequence. (->V0300) A036966: V0303: 3-full (cubefull) numbers. (See V8 for "squareful" numbers.) A036967: V0304: 4-full (tesseractfull) numbers. A069492: V0305: 5-full numbers. A069493: V0306: 6-full numbers. — : V0307: 7-full numbers. — : V0308: 8-full numbers. A375975: V0309: Minimal θ(m) for m that is a product of a prime n-tuple. {1, 2, 5, 15, 53, 188, 691, 2547, 9551, 36155, 137887, 527455, 2021002, 7764393, ...} A244974: • V0310: ∑({ m ≤ n : rad(m) | n }). A243103: • V0311: ∏({ m ≤ n : rad(m) | n }) = ∏_{k=1..n} k^( ⌊n^k/k⌋ − ⌊(n^k -1)/k⌋ ). A301892: • V0312: θ(P(n)) = A010846(A002110(n)). A067255: V0320: “multiplicity notation”: exponents of primes p | n. Abbreviated MN(n). (See also A054841, a big-endian decimal concatenation of A067255.) A287352: • V0321: “π-code” (pi-code), first differences of exponents of the prime divisors p arranged in order of magnitude of p from least to greatest. Abbreviated PC(n). — : V0322: “zero-code”: ZC(A288813(n)), row n contains runs of zeros delimited by nonzero terms in A054841(A288813(n)), eliminating leading zeros. This assigns a “family code” to terms in A288813. Equivalent to subtracting 1 from every term in A067255(n) and eliminating leading zeros. A304886: • V0323: “primorial-code”: PR(A025487(n)), row n contains the indices of primorials pn# that together produce A025487(n). A124832: V0324: MN ↦ A025487. A079277: V0330: Largest n-regular k < n. A289280: V0331: Smallest n-regular k > n. (Always semidivisor). A301893: V0332: Numbers k that set records in θ(k)/τ(k), i.e., A010846(k)/A5(k). A275581: • V0333: Numbers n such that θ(n) ≥ n/2. A113839: V0334: Powerful sandwiched between twin primes. (->V0308->V0310) A076445: V0335: Smaller of twin powerful numbers. (->V0311) A369609: • V0341: Row n is list of n-coregular k <= n. A293780: V0342: Highly coregular numbers. — : V0350: Minimal θ(k) for k of a given prime constellation (A186634) such that ω(k) = n. {1, 2, 5, 15, 55, 192, 715, 2547, 9551, 36155, 137887, 527455, 2021002, ...} — : V0351: A138109: V0352: ϰ such that ω(ϰ) = 2 and θ(ϰ) = 5. A363061: • V0360: θ(P(n)) = A010846(A2110(n)). A010709: V0400: {4} = V4 ∩ V12 = A246547 ∩ V2182. A000079: V0401: Powers of 2. A000244: V0402: Powers of 3. A000351: V0403: Powers of 5. A000420: V0404: Powers of 7. A001020: V0405: Powers of 11. A001022: V0406: Powers of 13. A001026: V0407: Powers of 17. A001029: V0408: Powers of 19. A009967: V0409: Powers of 23. A030078: V0410: Prime cubes p^3. A030514: V0411: p^4. A050997: V0412: p^5. A030514: V0413: p^6. — : V0414: p^7. A179645: V0415: p^8. A030629: V0416: p^10. A030631: V0417: p^12. A083722: V0418: Product of n-nondivisor primes q : gpf(n) < q < n. A083720: V0419: Product of n-nondivisor primes q < gpf(n). — • V0420: V4 \ {V120 ∪ V401} = A246547 \ {A79 ∩ A1248}. (->V0490) A244623: V0421: V4 \ V401 = A246547 \ A79. (->V0491) A066760: V0491 Neutral sum function ξ(1,n) = 1 + n × (n+1)/2 − σ(n) − n × ϕ(n)/2. — : • V0492 Indices of records in V0491. — : • V0493 Records in V0491. — : • V0494 Neutral product function. — : • V0495 Indices of records in V0494. — : • V0496 Records in V0494. A019565: V0500: lex order squarefree numbers, factors written in decreasing order. — : • V0510: Decimal equivalent of the binary number formed by {f(4n+3), f(4n+2), f(4n+1)}, where f(x) = [x is squarefree] A366882: • V0511: A5117 \ A2110. A004394: V0512: superabundant numbers. A365435: • V0513: Position of primorials among squarefree numbers. A004490: V0514: colossally abundant numbers. A073751: V0515: A004490(n) is the product of first n terms of this sequence. A062503: V0520: Squarefree numbers squared. A072777: V0522: Powers of squarefree numbers that are not squarefree. A374000: • V0525: Products of the smallest instance of prime constellation A186634 starting with A186702. A258332: V0530: Position of 0s in V0610: m such that none of (4m + 1), (4m + 2), and (4m + 3) are squarefree. — : • V0531: Position of 1s in V0510 — : • V0532: Position of 2s in V0510 — : • V0533: Position of 3s in V0510 — : • V0534: Position of 4s in V0510 — : • V0535: Position of 5s in V0510 — : • V0536: Position of 6s in V0510 — : • V0537: Position of 7s in V0510: m such that all of (4m + 1), (4m + 2), and (4m + 3) are squarefree. — : • V0538: m such that 2 consecutive terms in (4m + 1), (4m + 2), and (4m + 3) are squarefree, but not all are such. — : • V0539: m such that (4m ± 1) are squarefree. — : • V0540: Position of {4, 1} in V0510: m such that only (4m ± 1) are squarefree, and (4m ± k), k = 2..3 are not. — : • V0541: Position of k in {1, 4, 9} in V0610: m such that only (4m + 1), (4m + 2), or (4m + 3) is squarefree. — : • V0542: Position of k in {3, 5, 6} in V0610: m such that only 2 of (4m + 1), (4m + 2), and (4m + 3) are squarefree. — : V0571: { ϰq : q = A053669(ϰ), ϰ ∈ A5117 } = A087560(A5117(n)). — : • V0591 Semidivisor sum function ξ_d(1,n). — : • V0592 Indices of records in V0591. — : • V0593 Records in V0591. — : • V0594 Semidivisor product function. — : • V0595 Indices of records in V0594. — : • V0596 Records in V0594. A010722: V0600: {6} = V6 ∩ V12 = V0111 ∩ V12 = A120944 ∩ A2182 = A2110 ∩ A2182. A350352: V0601: V6 \ V103 = A120944 \ A6881. A177492: V0602: A120944² = V6². A177493: V0603: A120944³ = V6³. — : • V0604: A120944⁴ = V6⁴. — : • V0605: A020639 ↦ A120944 = lpf(A120944(n)) = V1001 ↦ V6. A376833: • V0606: A119288 ↦ A120944 = slpf(A120944(n)) = V1002 ↦ V6. — : • V0607: A006530 ↦ A120944 = gpf(A120944(n)) = V1003 ↦ V6. A161620: V0608: Oblong primorials; ⌊√P(i)⌋ | P(i) for some i. A070195: • V0609: Varius sandwiched between twin primes. — : • V0610: Decimal equivalent of the binary number formed by {f(4n+3), f(4n+2), f(4n+1)}, where f(x) = [Omega(x) = omega(x) > 1] A046388: V0611: Odd squarefree semiprimes. A100484: V0612: Even squarefree semiprimes. A007304: V0613: Sphenic numbers. A046386: V0614: Products of 4 distinct primes. A046387: V0615: Products of 5 distinct primes. — : V0616: Products of 6 distinct primes. — : V0617: Products of 7 distinct primes. — : V0618: Products of 8 distinct primes. — : V0619: Products of 9 distinct primes. A039956: V0620: Even varius numbers = {A039956 \ {2}}. A024556: V0621: Odd varius numbers. — : • V0622: Varius divisible by 6. A367018: • V0623: Varius indivisible by 6. — : • V0624: Varius divisible by 30. — : • V0625: Varius indivisible by 30. — : • V0626: Varius divisible by 210. — : • V0627: Varius indivisible by 210. — : • V0628: Varius divisible by 2310. — : • V0629: Varius indivisible by 2310. — : • V0630: Position of 0s in V0610: m such that none of (4m + 1), (4m + 2), and (4m + 3) are squarefree and composite. — : • V0631: Position of 1s in V0610 — : • V0632: Position of 2s in V0610 — : • V0633: Position of 3s in V0610 — : • V0634: Position of 4s in V0610 — : • V0635: Position of 5s in V0610 — : • V0636: Position of 6s in V0610 — : • V0637: Position of 7s in V0610: m such that all of (4m + 1), (4m + 2), and (4m + 3) are squarefree and composite. — : • V0638: m such that 2 consecutive terms in (4m + 1), (4m + 2), and (4m + 3) are squarefree and composite, but not all are such. — : • V0639: m such that (4m ± 1) are squarefree and composite. — : • V0640: Position of {4, 1} in V0610: m such that only (4m ± 1) are squarefree and composite, and (4m ± k), k = 2..3 are not. — : • V0641: Position of k in {1, 4, 9} in V0610: m such that only (4m + 1), (4m + 2), or (4m + 3) is squarefree and composite. — : • V0642: Position of k in {3, 5, 6} in V0610: m such that only 2 of (4m + 1), (4m + 2), and (4m + 3) are squarefree and composite. A294576: • V0649: semitotative-dominant numbers (2 ×ξt (n) > n). Products of prime constellations: A037074: V0650: Products of a pair of twin primes (Cf. V1500). A372319: • V0651: (k=3, d = {0,2,6}) Products of prime triples with spacing d (Cf. V1501). A372419: • V0652: (k=3, d = {0,4,6}) Products of prime triples with spacing d (Cf. V1502). A138637: V0653: (k=4, d = {0,2,6,8}) Products of prime quadruples with spacing d (Cf. V1503). A375263: • V0654: (k=5, d = {0,2,6,8,12}) Products of prime quintuples with spacing d (Cf. V1504). A375264: • V0655: (k=5, d = {0,4,6,10,12}) Products of prime quintuples with spacing d (Cf. V1505). A375418: • V0656: (k=6, d = {0,4,6,10,12,16}) Products of prime sextuples with spacing d (Cf. V1506). A375644: • V0657: (k=7, d = {0,2,6,8,12,18,20}) Products of prime septuples with spacing d (Cf. V1507). A375645: • V0658: (k=7, d = {0,2,8,12,14,18,20}) Products of prime septuples with spacing d (Cf. V1508). A375646: • V0659: (k=8, d = {0,2,6,8,12,18,20,26}) Products of prime octuples with spacing d (Cf. V1509). A375647: • V0660: (k=8, d = {0,2,6,12,14,20,24,26}) Products of prime octuples with spacing d (Cf. V1510). A375648: • V0661: (k=8, d = {0,6,8,14,18,20,24,26}) Products of prime octuples with spacing d (Cf. V1511). — : • V0662: (k=9, d = {0,2,6,8,12,18,20,26,30}) (Cf. V1512, A022545). — : • V0663: (k=9, d = {0,2,6,12,14,20,24,26,30}) (Cf. V1513, A022546). — : • V0664: (k=9, d = {0,4,6,10,16,18,24,28,30}) (Cf. V1514, A022547). — : • V0665: (k=9, d = {0,4,10,12,18,22,24,28,30}) (Cf. V1515, A022548). — : • V0666: (k=10, d = {0,2,6,8,12,18,20,26,30,32}) (Cf. V1516, A027569). — : • V0667: (k=10, d = {0,2,6,12,14,20,24,26,30,32}) (Cf. V1517, A027570). — : • V0668: (k=11, d = {0,2,6,8,12,18,20,26,30,32,36}) (Cf. V1518, A213647). — : • V0669: (k=11, d = {0,4,6,10,16,18,24,28,30,34,36}) (Cf. V1519, A213646). — : • V0670: (k=12, d = {0,2,6,8,12,18,20,26,30,32,36,42}). (Cf. V1520, A213645). — : • V0671: (k=12, d = {0,6,10,12,16,22,24,30,34,36,40,42}). (Cf. V1521, A213601). — : • V0672: (k=13, d = {0,2,6,8,12,18,20,26,30,32,36,42,48}). (Cf. V1522, A257139). — : • V0673: (k=13, d = {0,4,6,10,16,18,24,28,30,34,40,46,48}). (Cf. V1523, A257137). — : • V0674: (k=13, d = {0,4,6,10,16,18,24,28,30,34,36,46,48}). (Cf. V1524, A257138). — : • V0675: (k=13, d = {0,2,8,14,18,20,24,30,32,38,42,44,48}). (Cf. V1525, A257140). — : • V0676: (k=13, d = {0,2,12,14,18,20,24,30,32,38,42,44,48}). (Cf. V1526, A257141). — : • V0677: (k=13, d = {0,6,12,16,18,22,28,30,36,40,42,46,48}). (Cf. V1527, A214947). — : • V0678: (k=14, d = {0,2,6,8,12,18,20,26,30,32,36,42,48,50}). (Cf. V1528, A257167). — : • V0679: (k=14, d = {0,2,8,14,18,20,24,30,32,38,42,44,48,50}). (Cf. V1529, A257168). (See V670_ for more) A138109: V0680: Semiprimes p × q, p < q < p². (->V0670) — : • V0681: { qϰ : q = A053669(ϰ), ϰ ∈ A120944 } = A087560(A120944(n)). (->V0671) — : • V0682: { p₂ϰ : p₂ = A119288(ϰ), ϰ ∈ A120944 }. (->V0672) — : • V0683: { max(p₂,q) × ϰ : p₂ = A119288(ϰ), q = A053669(ϰ), ϰ ∈ A120944 }. (->V0673) — : • V0684: Varius sandwiched between prime powers {6, 10, 26, 30, 42, 82, 102, 138, ...}. A070195: V0685: Varius sandwiched between primes {6, 30, 42, 102, 138, 282, 462, 570, 618, ...}. A073251: V0686: First number in varius triples {33, 85, 93, 141, 185, 201, 213, 217, 253, ...}. — : • V0687: Varius sandwiched between tantus {51, 55, 91, 161, 235, 249, 295, 305, ...}. — : • V0688: Varius sandwiched between powerful {26, 70226, 189750625, 512706121225, ...}. A138511: V0689: Semiprimes p × q, p < q, p² < q. A366413: • V0690 = V6 \ V111 = A120944 \ A2110. (->V0602) — : V0691 = V601 \ V111 = V602 \ V103 = V6 \ {V103 ∪ V111}. (->V0603) A360768: • V0700: Strong tantus numbers, i.e., tantus that foster k ⑨ n. (->V0070) A360767: • V0701: Weak tantus numbers, A126708 \ A360768 = V7 \ V0702. (->V0071) A360765: • V0702: Thick tantus numbers, i.e., tantus that foster k ③ n. (->V0072) A363082: • V0703: Thin tantus numbers, A126708 \ A363082 = V7 \ V0704. (->V0073) A363814: • V0704: A126706 ∩ A055932 = V7 ∩ V0210. (->V0730) A056808: V0705: A126706 ∩ A056808 = V7 ∩ V0212 A364710: • V0706: A126706 ∩ A025487 = V7 ∩ V0211. (->V0732->V0705) — : • V0707: A126706 ∩ A002182 = V7 ∩ V12. A367268: • V0708: A126706 \ A025487 = V7 \ V0211. (->V0736) A368089: • V0709: A126706 \ A055932 = A126706 ∩ A080259 = V7 \ V0210 = V7 ∩ V0219. (->V0735) (— : • V0706: A126706 ∩ A363280 = A126706 ∩ {A025487 \ A002182}. (->V0733)) — : • V0710: A366825 ∩ A055932 = V70 ∩ V0210. — : • V0711: A366825 ∩ A056808 = V70 ∩ V0212. — : • V0712: A366825 ∩ A025487 = V70 ∩ V0211. — : • V0713: A366825 ∩ A363280 = V70 ∩ {V0211 \ V12}. — : • V0714: A366825 ∩ A002182 = V70 ∩ V12. — : • V0715: A366825 ∩ A333655 = V70 ∩ {V12 \ V14}. — : • V0716: A366825 ∩ A002201 = V70 ∩ V14. — : • V0717: A126706 \ A366825. — : • V0718: A366825 \ A025487 = V70 \ V0211. — : • V0719: A366825 \ A055932 = V70 \ V0210. — : • V0720: A303946 ∩ A055932 = V72 ∩ V0210. — : • V0721: A303946 ∩ A056808 = V72 ∩ V0212. — : • V0722: A303946 ∩ A025487 = V72 ∩ V0211. — : • V0723: A303946 ∩ A363280 = V72 ∩ {V0211 \ V12}. — : • V0724: A303946 ∩ A002182 = V72 ∩ V12. — : • V0725: A303946 ∩ A333655 = V72 ∩ {V12 \ V14}. — : • V0726: A303946 ∩ A002201 = V72 ∩ V14. — : • V0727: A366250 ∩ A002182 = V3302 ∩ V12 = {48, 45360, 50400} = A367511 \ {1, 4, 36}. (->V3324) — : • V0728: A303946 \ A025487 = V72 \ V0211. — : • V0729: A303946 \ A055932 = V72 \ V0210. — : • V0730: A364702 ∩ A055932 = V73 ∩ V0210. — : • V0731: A364702 ∩ A056808 = V73 ∩ V0212. (->V0748). — : • V0732: A364702 ∩ A025487 = V73 ∩ V0211. (->V0747). — : • V0733: A364702 ∩ A363280 = V73 ∩ {V0211 \ V12}. — : • V0734: A364702 ∩ A002182 = V73 ∩ V12. — : • V0735: A364702 ∩ A333655 = V73 ∩ {V12 \ V14}. — : • V0736: A364702 ∩ A002201 = V73 ∩ V14. — : • V0737: A366250 ∩ A002182 = V3302 ∩ V12. = {48, 45360, 50400}. — : • V0738: A364702 \ A025487 = V73 \ V0211. (->V0745). — : • V0739: A364702 \ A055932 = V73 \ V0210. (->V0746). — : • V0740: A361098 ∩ A055932 = V74 ∩ V0210. (->V0742). — : • V0741: A361098 ∩ A056808 = V74 ∩ V0212. — : • V0742: A361098 ∩ A025487 = V74 ∩ V0211. (->V0743). — : • V0743: A361098 ∩ A363280 = A361098 ∩ {A025487 \ A002182} — : • V0744: A361098 ∩ A333655 = V74 ∩ {V12 \ V14}. — : • V0745: A361098 ∩ A6939 \ {1,2} = V74 ∩ V0112. — : • V0746: A361098 \ A6939 = V74 \ V0112. — : • V0747: A361098 \ A2182. — : • V0748: A361098 \ A025487 = V74 \ V0211. (->V0746) — : • V0749: A361098 \ A055932 = A361098 ∩ A080259 = V74 \ V0210 = V74 ∩ V0219. (->V0745) A088860: • V0750: A364999 ∩ A055932 = A364999 ∩ A025487 = V75 ∩ V0210 = V75 ∩ V0211 = {2 × P(2..∞)}. (->V0752, V0753). — : • V0751: A367708 ∩ A055932 = V3304 ∩ V0210. (->V3340) — : • V0752: A367708 ∩ A056808 = V3304 ∩ V0212. (->V3341) — : • V0753: A367708 ∩ A025487 = V3304 ∩ V0211. (->V3342) — : • V0754: A367708 ∩ A363280 = V3304 ∩ {V0211 \ V12}. (->V3343) — : • V0755: A367708 ∩ A002182 = V3304 ∩ V12. (->V3344) — : • V0756: A367708 ∩ A002201 = V3304 ∩ V14. (->V3346) — : • V0757: A367708 \ A025487 = V3304 \ V0211. (->V3348) — : • V0758: A367708 \ A055932 = V3304 \ V0210. (->V3349) — : • V0759: A364999 \ A055932 = A364999 \ A025487 = V75 \ V0210 = V75 \ V0211. A366250: • V3302: ∪ { k = m × rad(k) : ω(k) > 1, rad(m) | rad(k), k ≥ rad(k)² } ∪ {1}. A369540: • V0760: A364998 ∩ A055932 = V76 ∩ V0210 = { m × P(n) : 3 ≤ m < q, n ≥ 2 } ⊂ A060735. (->V0762). A369419: • V0761: A364998 ∩ A056808 = V76 ∩ V0212 = { m × P(n) : 3 ≤ m < q, n ≥ 2, m ∉ A025487 } A369541: • V0762: A364998 ∩ A025487 = V76 ∩ V0211 = { m × P(n) : 3 ≤ m < q, n ≥ 2, m ∈ A025487 }. (->V0763). — : • V0763: A366250 ∩ A055932 = V3302 ∩ V0210. (->V3320) — : • V0764: A366250 ∩ A056808 = V3302 ∩ V0212. (->V3321) — : • V0765: A366250 ∩ A025487 = V3302 ∩ V0211. (->V3322) — : • V0766: A366250 \ A025487 = V3302 \ V0211. (->V3328) — : • V0767: A366250 \ A055932 = V3302 \ V0210. (->V3329) — : • V0768: A364998 \ A025487 = V76 \ V0211. — : • V0769: A364998 \ A055932 = V76 \ V0210. A369150: • V0770: Nonminimally weak-thick tantus = V77 \ V0771. (->V0706) A366460: • V0771: Odd minimally tantus numbers { k = p₁ × ϰ : p₁ = lpf(ϰ) > 2, ϰ = rad(k) }. (->V0071) A010851: • V0772: {12} = V0112 ∩ V75. — : • V0773: {60}. — : • V0774: {360} = V74 ∩ V0113 ∩ V14. (->V0744) — : • V0775: V75 ∩ V14 = V75 ∩ V12 = V0701 ∩ V14. {12, 60}. (-V0753>) — : • V0776: V76 ∩ V14 = V778 ∩ V14 = {24, 120, 180, 840, 1260, 1680, 27720}. (->V0763) — : • V0777: V76 ∩ V12 = {120}. (->V0766) — : • V0778: V78 ∩ V12 = {12, 24, 60, 120, 180, 840, 1260, 1680, 27720} = A168263 \ {1, 2, 4, 6} = V0118 \ {1, 2, 4, 6}. (->V0708) — : • V0779: {36} = A286708 ∩ A2182 = V8 ∩ V12 = V80 ∩ V12 = V81 ∩ V12. (->V0800) — : • V0780: A364996 ∩ A055932 = V78 ∩ V0210. (->V0782) — : • V0781: A364996 ∩ A056808 = V78 ∩ V0212. — : • V0782: A364996 ∩ A025487 = V78 ∩ V0211. (->V0783). — : • V0783: A364996 ∩ A363280 = V78 ∩ {V0211 \ V12}. — : • V0784: A364996 ∩ A002182 = V78 ∩ V12. = {12, 24, 60, 120, 180, 840, 1260, 1680, 27720} = A168263 \ {1, 2, 4, 6} = V0118 \ {1, 2, 4, 6}. (->V0708) — : • V0785: A364996 ∩ A333655 = V78 ∩ {V12 \ V14}. — : • V0786: A364996 ∩ A002201 = V78 ∩ V14. — : • V0787: — : • V0788: A364996 \ A025487 = V75 \ V0211. — : • V0789: A364996 \ A055932 = V75 \ V0210. — : • V0790: A332785 ∩ A055932 = V79 ∩ V0210. — : • V0791: A332785 ∩ A056808 = V79 ∩ V0212. — : • V0792: A332785 ∩ A025487 = V79 ∩ V0211. — : • V0793: A332785 ∩ A363280 = {A126706 \ A286708} ∩ {A025487 \ A002182} — : • V0794: A332785 ∩ A2182 = V79 ∩ V12. — : • V0795: A332785 ∩ A333655 = V79 ∩ {V12 \ V14}. — : • V0796: A332785 ∩ A2201 = V79 ∩ V14. — : • V0797: A332785 \ A2182 = V79 \ V12. — : • V0798: A332785 \ A025487 = V79 \ V0211. — : • V0799: A332785 \ A055932 = V79 \ V0210. A369374: • V0800: A286708 ∩ A055932 = V8 ∩ V0210. (->V0830->V0801) A369420: • V0801: A286708 ∩ A056808 = V8 ∩ V0212. (->V0831->V0802) A364930: • V0802: A286708 ∩ A025487 = V8 ∩ V0211. (->V0832->V0803) A372695: • V0803: 3-pithy numbers þ(3); cubefull plenus numbers. A372841: • V0804: 4-pithy numbers þ(4); tesseractfull plenus numbers. — : • V0805: 5-pithy numbers þ(5). — : • V0806: 6-pithy numbers þ(6). — : • V0807: 7-pithy numbers þ(7). A369636: • V0808: A286708 \ A025487 = V8 \ V0211. (->V0834->V0805) A369417: • V0809: A286708 \ A055932 = V8 \ V0210. (->V0833->V0804) — : • V0810: A131605 ∩ A055932 = V81 ∩ V0210. (->V0840) — : • V0811: A131605 ∩ A056808 = V81 ∩ V0212. (->V0841) A368682: • V0812: A131605 ∩ A025487 = V81 ∩ V0211. (->V0842) — : • V0813: A131605 ∩ A368507 = V81 ∩ V0113. (->V0845) A372404: • V0814: k ∈ A286708 : k/rad(k) ∈ A013929. Plenus with a nonsquarefree kernel ratio A177492: V0815: k ∈ A286708 : k/rad(k) ∈ A005117. Plenus with a squarefree kernel ratio — : • V0816. — : • V0817: A131605 \ A368507 = V81 \ V0113. (->V0846) — : • V0818: A131605 \ A025487 = V81 \ V0211. (->V0844) — : • V0819: A131605 \ A055932 = V81 \ V0210. (->V0843) — : • V0820: A366854 ∩ A055932 = V82 ∩ V0210. (->V0860) — : • V0821: A366854 ∩ A056808 = V82 ∩ V0210. (->V0861) — : • V0822: A366854 ∩ A025487 = V82 ∩ V0211. (->V0862) — : • V0823: A366854 ∩ A368507 = V82 ∩ V0113. (->V0864) — : • V0824: A366854 ∩ A025487 \ A368507 = V82 ∩ V0211 \ V0113. (->V0863) — : • V0825. — : • V0826. — : • V0827: A366854 \ A368507 = V82 \ V0113. (->V0867) — : • V0828: A366854 \ A025487 = V82 \ V0211. (->V0866) — : • V0829: A366854 \ A055932 = V82 \ V0210. (->V0865) A303606: V0080: Planus numbers, varius numbers at multiplicity: { v^δ : v ∈ A120944, δ > 1 }. A131605: V0081: Perfect-power plenus (valens) numbers, A286708 ∩ A1597 = V8 ∩ V2 = V81 ∪ V86. (->V84) A366854: • V0082: Fortis numbers, tantus numbers at multiplicity: { t^δ : t ∈ A126706, δ > 1 } = A131605 \ A303606 = V81 \ V80. (->V83) A365308: • V0083: Perfect powers of composite primorials, primorials at multiplicity, A100778 ∩ A303606. (->V81) — : • V0084: — : • V0085: — : • V0086: A365745: • V0087: A303606 \ A365308 = V80 \ V81. (->V0839->V0809) A052486: • V0088: Achilles numbers, A286708 \ A1597 = V8 \ V81 = V8 \ V2. (->V85) A359280: • V0089: Citroque numbers: V8 \ V80 = A286708 \ A303606. — : • V0880: A052486 ∩ A055932 = V88 ∩ V0210. (->V0850) — : • V0881: A052486 ∩ A056808 = V88 ∩ V0212. (->V0851) — : • V0882: A052486 ∩ A025487 = V88 ∩ V0211. (->V0852) — : • V0883. — : • V0884. — : • V0885. — : • V0886. — : • V0887. — : • V0888: A052486 \ A025487 = V85 \ V0211. (->V0854) — : • V0889: A052486 \ A055932 = V88 \ V0210. (->V0853) — : • V0890: A359280 ∩ A055932 = V89 ∩ V0210. — : • V0891: A359280 ∩ A056808 = V89 ∩ {V0210 \ V0211}. — : • V0892: A359280 ∩ A025487 = V89 ∩ V0211. A368508: • V0893: A359280 ∩ A368507 = V89 ∩ V0113 = V89 ∩ V0113. (A6939(n)^k, k > 1, n > 1). (->V0895) — : • V0894: A359280 ∩ A025487 \ A368507 = V82 ∩ V0211 \ V0113. (->V0897) — : • V0895: — : • V0896: — : • V0897: A359280 \ A368508 = V89 \ V0895. (->V0896) — : • V0898: A359280 \ A025487 = V89 \ V0211. (->V0894) — : • V0899: A359280 \ A055932 = V89 \ V0210. (->V0893) A001223: V1000: Prime gaps: a(n) = prime(n+1) - prime(n). A020639: V1001: Least prime divisor of n, lpf(n). A119288: V1002: Second least distinct prime divisor of non-prime-power n. A006530: V1003: Greatest prime divisor of n, gpf(n). A087560: V1009: { nq : q = A053669(ϰ) }. (->V1011) A365535: • V1010: Composite numbers k such that the core and the kernel of k are equal. A275055: • V1020 list of divisors d of n in order of appearance in a matrix of products that arranges the powers of prime divisors p of n along independent axes. — : • V1021: A091050: V1022: Number of divisors d | n that are perfect powers. (Posed, recycled as A376697) A183094*: V1023: Number of divisors d | n such that rad(d)² | m. * excludes 1. A046660: V1024: Number of divisors d | n that are prime powers p^m. A034444: V1025: Number of squarefree d | n. (u-tau). A376504: • V1026: Number of composite squarefree divisors d | n. A376514: • V1027: Number of divisors d | n that are neither squarefree nor prime powers. — : • V1028: Number of divisors d | n that are not prime powers such that rad(d)² | d. A051681: V1029: a(n) is the least of n consecutive nonsquarefree terms. A376271: • V1047: Numbers m such that V1027(m) > 1. Subset of V7 \ V70. Right and Left Gauntlet Primes. — : • V1070: {0, 1, 1, 2, 30, 457, 1831, 59257, 560634, 2279181, ...} — : • V1071: {0, 1, 1, 1, 28, 260, 2806, 34210, 106280, 7774333, ...} — : • V1072: — : • V1073: k=3. {2, 4, 6, 9, 11, 12, 15, 16, 18, 19, ...} — : • V1074: k=4. {30, 34, 42, 47, 62, 68, 82, 115, 141, 146, 150, ...} — : • V1075: k=5. {457, 462, 641, 804, 876, 890, 1051, ...} — : • V1076: k=6. {1831, 2788, 3077, 3644, 4522, 4743, 5949, 7393, ...} — : • V1077: k=7. {59257, 70331, 80413, 90953, 102703, 111420, ...} — : • V1078: k=8. {560634, 696197, 1036796, 1337806, 1514187, 1736516, ...} — : • V1079: k=9. {2279181, 7391990, 10656643, 14350249, ...} First prime of k consecutive primes whose product m where only 1 prime p is such that log_p m > k. A374873: • V1080: Smallest p_1 where products m of n primes p_1..p_n are such that only p_1 < m^(1/n). — : • V1081: Smallest p_1 where products m of n primes p_1..p_n are such that only p_n > m^(1/n). — : • V1082: A372209: • V1083: k=3. {3, 7, 13, 23, 31, 37, 47, 53, 61, 67, 73, 89, ...} A375974: • V1084: k=4. {113, 139, 181, 211, 293, 337, 421, 631, ...} A376136: • V1085: k=5. {3229, 3271, 4759, 6173, 6803, 6917, 8389, 8971, ...} — : • V1086: k=6. {15683, 25261, 28229, 34061, 43331, 45779, 58831, ...} — : • V1087: k=7. {736279, 886913, 1025957, 1172579, 1337911, 1461211, ...} — : • V1088: k=8. {8332427, 10509371, 16094621, 21132383, 24120233, ...} — : • V1089: k=9. {37305713, 130259693, 191935769, 262985953, 335319071, ...} A183093: V1101: 1 less than Lean Divisor Counting Function τ₄(n). A183094: V1102: 1 less than Plenary Divisor Counting Function τ₆(n). A000012: V1111: {1} A352475: • V1120: Numbers m such that τ(m) ⊥ 6. A350014: • V1121: Numbers m such that τ(m²) ⊥ 6. A354178: V1122: Numbers m such that τ(m) ⊥ 30. A354178: V1122: Numbers m such that τ(m²) ⊥ 30. A358001: • V1124: Numbers m such that τ(m) ⊥ 210. A358250: • V1125: Numbers m such that τ(m²) ⊥ 210. — : • V1126: Numbers m such that τ(m) ⊥ 2310. — : • V1127: Numbers m such that τ(m²) ⊥ 2310. — : • V1128: Numbers m such that τ(m) ⊥ 30030. — : • V1129: Numbers m such that τ(m²) ⊥ 30030. A355058: • V1130: Numbers m such that τ(m) mod 6 ≡ 3. A354799: • V1131: m ∈ A1694 such that 3 | τ(m²), where τ(n) = A5(n). A372972: • V1132: Numbers n such that τ(n) < Θ(n), A5(n) < A8479(n). A372864: • V1133: Numbers n such that τ(n) = Θ(n), A5(n) = A8479(n). A372720: • V1134: τ(n) − Θ(n) = A5(n) − A8479(n). A373737: • V1135: Maximal j such that τ(S_ϰ(i)) − i > 0 for ϰ such that Ω(ϰ) = ω(ϰ) > 1 and i ≤ j. — : • V1136: Maximal i such that τ(S_ϰ(i)) − i > 0 for ϰ such that Ω(ϰ) = ω(ϰ) > 1. — : • V1137: | { i : τ(S_ϰ(i)) − i > 0, Ω(ϰ) = ω(ϰ) > 1} |. — : • V1138: S_ϰ(j) for maximal j : τ(S_ϰ(i)) − i > 0, Ω(ϰ) = ω(ϰ) > 1, i ≤ j. — : • V1139: A371630: • V1140: Numbers that set records in V1134. A371634: • V1141: Records in V1134. On the largest primorial divisor p_ω(m)# of highly composite and superabundant m: A340840: • V1200: Highly composite or superabundant numbers: union of A2182 and A4394. A166981: V1201: Superabundant numbers that are highly composite. A224078: V1202: Superior highly composite numbers that are colossally abundant. (finite: 449 terms) A304234: • V1203: Superior highly composite numbers that are superabundant but not colossally abundant. (39 terms) A304235: • V1204: Colossally abundant numbers that are highly composite, but not superior highly composite. (34 terms) A338786: • V1205: Numbers in A166981 that are neither superior highly composite nor colossally abundant. A365901: • V1206: Irr. tri. read by rows giving trajectory beginning with A002182(n) under recursion of x −> f(x) until reaching 1, where f(x) = x/rad(x), rad(x) = A007947(x). A365710: • V1207: A119288(A126706(n)) = V1002 ↦ V7. A365900: • V1208: HCNs k that remain highly composite when recursively divided by squarefree kernel. A365902: • V1209: Irr. Tri. h(n) = A002182(n) arranged first according to rad(h(n))/h(n) then by rad(h(n)) A212182: V1210: A067255(A2182(n)). (Flammenkamp coded) A306737: • V1211: Indices k of primorials P(k) whose product is A2182(n)) (Noe coded). — : • V1212: A067255(A4394(n)). (Flammenkamp coded) A307322: • V1213: Indices k of primorials P(k) whose product is A4394(n)) (Noe coded). A098719: V1220: Indices k such that A025487(k) = A2110(n). A306802: • V1221: Indices k such that A025487(k) = A2182(n). — : • V1222: Indices k such that A025487(k) = A4394(n). — : • V1223: — : • V1224: A331119: • V1225: Indices k such that A055932(k) = A025487(n). A331938: • V1226: Indices k such that A055932(k) = A2110(n). A332034: • V1227: Indices k such that A055932(k) = A2182(n). A332035: • V1228: Indices k such that A055932(k) = A4394(n). A332241: • V1229: Indices k such that A055932(k) = A224078(n). A108602: V1230: ω(m) for highly composite m (in A2182). A305025: • V1231: ω(m) for superabundant m (in A4394). A307113: • V1232: Number of HCNs m in the interval P(n) ≤ m < P(n+1). A307327: • V1233: Number of superabundant m in the interval P(n) ≤ m < P(n+1). A301413: • V1240: A002182(n)/A002110(A108602(n)), i.e., m/pω(m)# for highly composite m. A301414: • V1241: primitive values in A301413. A301415: • V1242: Number of terms m in A002110 such that A301413(k) × A002110(m) is in A002182. A305056: • V1245: A004394(n)/A002110(A001221(A004394(n))), i.e., m/pω(m)# for superabundant m. A340014: • V1246: primitive values in A305056. — • V1250: Union of A1241 and A1242 (A301414 ∪ A340014) (data). — • V1300: Union of A2201 and A4490 (data). A301416: • V1341: Numbers in A301413 that produce superior highly composite numbers when multiplied by some primorial. A340137: • V1342: Numbers in A305056 that produce colossally abundant numbers when multiplied by some primorial. — • V1350: Union of A1341 and A1342 (A301416 ∪ A340137) (data). A032742: V1400: A002201(n) is the product of first n terms of this sequence. A073751: V1411: Largest d < n such that d | n and d ≠ n. Prime constellations: A001359: V1500: (k=2, d = {0,2}) Lesser of twin primes. A022004: V1501: (k=3, d = {0,2,6}) Initial members of prime triples. A022005: V1502: (k=3, d = {0,4,6}) Initial members of prime triples. A007530: V1503: (k=4, d = {0,2,6,8}) Initial members of prime quadruples. A022006: V1504: (k=5, d = {0,2,6,8,12}) Initial members of prime quintuples. A022007: V1505: (k=5, d = {0,4,6,10,12}) Initial members of prime quintuples. A022008: V1506: (k=6, d = {0,4,6,10,12,16}) Initial members of prime sextuples. A022009: V1507: (k=7, d = {0,2,6,8,12,18,20}) Initial members of prime septuples. A022010: V1508: (k=7, d = {0,2,8,12,14,18,20}) Initial members of prime septuples. A022011: V1509: (k=8, d = {0,2,6,8,12,18,20,26}) Initial members of prime octuples. A022012: V1510: (k=8, d = {0,2,6,12,14,20,24,26}) Initial members of prime octuples. A022013: V1511: (k=8, d = {0,6,8,14,18,20,24,26}) Initial members of prime octuples. A022545: V1512: (k=9, d = {0, 2, 6, 8, 12, 18, 20, 26, 30}). A022546: V1513: (k=9, d = {0, 2, 6, 12, 14, 20, 24, 26, 30}). A022547: V1514: (k=9, d = {0, 4, 6, 10, 16, 18, 24, 28, 30}). A022548: V1515: (k=9, d = {0, 4, 10, 12, 18, 22, 24, 28, 30}). A027569: V1516: (k=10, d = {0, 2, 6, 8, 12, 18, 20, 26, 30, 32}). A027570: V1517: (k=10, d = {0, 2, 6, 12, 14, 20, 24, 26, 30, 32}). A213647: V1518: (k=11, d = {0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36}). A213646: V1519: (k=11, d = {0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 36}). A213645: V1520: (k=12, d = {0,2,6,8,12,18,20,26,30,32,36,42}). A213601: V1521: (k=12, d = {0,6,10,12,16,22,24,30,34,36,40,42}). A257139: V1522: (k=13, d = {0,2,6,8,12,18,20,26,30,32,36,42,48}). A257137: V1523: (k=13, d = {0,4,6,10,16,18,24,28,30,34,40,46,48}). A257138: V1524: (k=13, d = {0,4,6,10,16,18,24,28,30,34,36,46,48}). A257140: V1525: (k=13, d = {0,2,8,14,18,20,24,30,32,38,42,44,48}). A257141: V1526: (k=13, d = {0,2,12,14,18,20,24,30,32,38,42,44,48}). A214947: V1527: (k=13, d = {0,6,12,16,18,22,28,30,36,40,42,46,48}). A257167: V1528: (k=14, d = {0,2,6,8,12,18,20,26,30,32,36,42,48,50}). A257168: V1529: (k=14, d = {0,2,8,14,18,20,24,30,32,38,42,44,48,50}). Second primes in a (right) gauntlet that incorporates prime k-tuplets. — : • V1571: V1083 ⋂ V1500. 3-prime gauntlet with greater prime factors that are twin primes. — : • V1572: V1084 ⋂ V1501. 4-prime gauntlet with greater prime factors that are prime triplets. — : • V1573: V1084 ⋂ V1502. 4-prime gauntlet with greater prime factors that are prime triplets. — : • V1574: V1085 ⋂ V1503. 5-prime gauntlet with greater prime factors that are prime 4-tuplets. — : • V1575: V1086 ⋂ V1504. 6-prime gauntlet with greater prime factors that are prime 5-tuplets. — : • V1576: V1086 ⋂ V1505. 6-prime gauntlet with greater prime factors that are prime 5-tuplets. — : • V1577: V1087 ⋂ V1506. 7-prime gauntlet with greater prime factors that are prime 6-tuplets. — : • V1578: V1088 ⋂ V1507. 8-prime gauntlet with greater prime factors that are prime 7-tuplets. — : • V1579: V1088 ⋂ V1507. 8-prime gauntlet with greater prime factors that are prime 7-tuplets. A307540: • V2100: Irregular triangle T(n, k) such that squarefree m with gpf(m) = prime(n) in each row are arranged according to increasing values of φ(m)/m = A076512(m)/A109395(m). A006094: V2101: Products of 2 successive primes. (m with the second greatest φ(m)/m in column x) A070826: V2102: pn#/2. (odd m with least φ(m)/m in column x) A325236: • V2103: Squarefree k such that φ(k)/k − ½ is positive and minimal for k with gpf(k) = prime(n). A307544: • V2105: T(n,k) = A087207(A307540(n,k)). (Binary encoding of A307540) A325237: • V2107: Squarefree k such that ½ − φ(k)/k is positive and minimal for k with gpf(k) = prime(n). A077017: V2108: (even m with least φ(m)/m in column x) A306237: • V2109: pn#/prime(n − 1). (m with the second least φ(m)/m in column x) A247451: V2110: Largest primorial factor of A025487(n). A307107: • V2111: A025487(n)/A247451(n). A005843: V2400: even numbers; A005408: V2401: odd numbers; A016945: V2402: multiples of 3 coprime to 6. A084967: V2403: multiples of 5 coprime to 30. A084968: V2404: multiples of 7 coprime to 210. A084969: V2405: multiples of 11 coprime to 2310. A084970: V2406: multiples of 13 coprime to 30030. — : V2407: multiples of 17 coprime to V0111(7). — : V2408: multiples of 19 coprime to V0111(8). — : V2409: multiples of 23 coprime to V0111(9). A007310: V2500: T6 = numbers m coprime to 6 = (2 × 3). 5-rough numbers. A045572: V2501: T10 = numbers m coprime to 10 = (2 × 5). A162699: V2502: T14 = numbers m coprime to 14 = (2 × 7). A229829: V2503: T15 = numbers m coprime to 15 = (3 × 5). A160545: V2504: T21 = numbers m coprime to 21 = (3 × 7). A235933: V2505: T35 = numbers m coprime to 35 = (5 × 7). A007775: V2600: T30 = numbers m coprime to 30 = (2 × 3 × 5). 7-rough numbers. A206547: V2601: T42 = numbers m coprime to 42 = (2 × 3 × 7). A235583: V2602: T70 = numbers m coprime to 70 = (2 × 5 × 7). A236206: V2603: T105 = numbers m coprime to 105 = (3 × 5 × 7). A008364: V2650: T210 = numbers m coprime to 210 = (2 × 3 × 5 × 7). 11-rough numbers. A008365: V2651: T2310 = numbers m coprime to 2310 = (2 × 3 × 5 × 7 × 11). 13-rough numbers. A008366: V2652: T30030 = numbers m coprime to 30030 = (2 × 3 × 5 × 7 × 11 × 13). 17-rough numbers. A166061: V2653: T510510 = numbers m coprime to 510510 = (2 × 3 × 5 × 7 × 11 × 13 × 17). 19-rough numbers. A166063: V2654: T9699690 = numbers m coprime to 9,699,690 = (2 × 3 × 5 × 7 × 11 × 13 × 17 × 19). 23-rough numbers. — : V2655: T(P(9)). 29-rough numbers. — : V2656: T(P(10)). 31-rough numbers. — : V2657: T(P(11)). 37-rough numbers. — : V2658: T(P(12)). 41-rough numbers. — : V2659: T(P(13)). 43-rough numbers. A002072: V3000: Smallest m such that for all k > m, either k or k+1 has a prime factor p > prime(n). Edge of possible adjacency of prime(n)-smooth numbers. — : • V3001: a(n) = ⌈C(2n,n)/2 + C(2n,n)/(2n)⌉. Approx. of least θ(k) for squarefree k and ω(k) = n. {1, 2, 5, 14, 44, 152, 539, 1962, 7240, 27012, 101616, 384782, 1464752, ...}. — : • V3002: — : • V3003: — : • V3004: — : • V3005: — : • V3006: — : • V3007: a(n) is a squarefree k such that the n-th prime constellation product has minimum theta. {1, 2, 5, 16, 15, 53, 188, 188, 691, 2589, 2547, ...} (prime k-tuplets sorted lexically) — : • V3008: V3009 - V0300. — : • V3009: a(n) = ⌈¾ × C(2n,n)⌉. Approximation of V0309. — : • V3010: list of m in h_{ϰ,ℓ}, the ω(n)-rank hypertenuse of the Haüy orthosimplex r_{ϰ,ℓ}. — : • V3011: β(n) = Content of h_{ϰ,ℓ}, the ω(n)-rank hypertenuse of the Haüy orthosimplex r_{ϰ,ℓ}. — : • V3012: indices of records in β(n). — : • V3013: records in β(n). — : • V3014 — : • V3015: list of m not in h_{ϰ,ℓ}. — : • V3016: ψ(n) = Content of r_{ϰ,ℓ} \ h_{ϰ,ℓ}. — : • V3017: indices of records in ψ(n). — : • V3018: records in ψ(n). — : • V3019: characteristic function of m in h_{ϰ,ℓ} among r_{ϰ,ℓ}. A275280: • V3020 list of n-regular 1 ≤ k ≤ n (such that that k | ne with e ≥ 0), in order of appearance in a matrix of products that arranges the powers of prime divisors p of n along independent axes. — : • V3021: A376719: • V3022: Number of perfect powers m ≤ n such that rad(m) | n. A046660: • V3023: Number of m ≤ n such that both rad(m) | n and rad(m)² | m. A361373: • V3024: Number of prime powers p^m ≤ n such that p | n. A034444: V3025: Number of squarefree d | n. (u-tau). A376504: • V3026: Number of composite squarefree m ≤ n such that rad(m) | n. NOT the same as A327517(210) != a(210). A376505: • V3027: Number of m ≤ n, with m neither squarefree nor prime power, such that rad(m) | n. — : • V3028: Number of m ≤ n that are not prime powers such that both rad(m) | n and rad(m)² | m. A000295: • V3035: V3026(V111) = V3026 → A2110 = A295 Eulerian Numbers = V3026(k), k ∈ A025487. — : • V3036: V3026(V6) = V3026 → A126706. — : • V3037: V3027(V7) = V3027 → A126706. — : • V3038: V3028(V7) = V3028 → A126706. — : • V3040: {m : rad(m) | n, Ω(m) ≤ Ω(n) }. (The "ideal simplex" in R_n). — : • V3041: card({m : rad(m) | n, Ω(m) ≤ Ω(n) }) = C(Ω(n) + ω(n), ω(n)). — : • V3042: ∑({m : rad(m) | n, Ω(m) ≤ Ω(n) }). — : • V3043: ∏({m : rad(m) | n, Ω(m) ≤ Ω(n) }) = rad(n)^C(Ω(n) + ω(n), ω(n)+1). — : • V3044: V3041(n) - V31(n) = C(Ω(n) + ω(n), ω(n)) - θ(n). — : • V3045: indices of maxima in V3044. — : • V3046: indices of minima in V3044. A376384: • V3047: Numbers m such that V3037(m) > 1. A376834: • V3048: Numbers m such that V3038(m) > 0. — : • V3049: — : • V3050: {m ≤ n : rad(m) | n, Ω(m) > Ω(n) } A376846: • V3051: card({m ≤ n : rad(m) | n, Ω(m) > Ω(n) }) — : • V3052: indices of records in V3051. — : • V3053: records in V3051. — : • V3054: Numbers k such that V3051(k) > 0. — : • V3055: {m ≤ n : rad(m) | n, Ω(m) ≤ Ω(n) }. — : • V3056: card({m ≤ n : rad(m) | n, Ω(m) ≤ Ω(n) }) — : • V3057: indices of records in V3056. — : • V3058: records in V3056. — : • V3059: — : • V3060: {m > n : rad(m) | n, Ω(m) ≤ Ω(n) } — : • V3061: card({m ≤ n : rad(m) | n, Ω(m) > Ω(n) }) — : • V3062: indices of records in V3061. — : • V3063: records in V3061. — : • V3064: Numbers k such that V3061(k) > 0. A362041: • V3100: PrevCoregular(n). A065642: V3101: NextCoregular(n). A360719: • V3102: PrevCoregular(n) → V7. A360529: • V3103: NextCoregular(n) → V0109. A366786: • V3105: NextCoregular(n) → V5. A366807: • V3106: NextCoregular(n) → V6. A365788: • V3107: A010846(n) - A8479(n). = V31(n) - V34(n). A373738: • V3108: ϑ(ϰ) = ⌊1/ω(n)! × ∏_{p|n} 1+log(n)/log(p) ⌋ "thetapure" function. A372192: • V3109: V3108(n) - V31(n). A375011: • V3110: V3109(n) - V34(n). A364225: • V3111: V31(V211) = θ → A025487. — : • V3112: — : • V3113: V31(V3) = θ → A1694. — : • V3114: A363924: • V3115: V31(V5) = θ → A5117. — : • V3116: V31(V6) = θ → A120944. — : • V3117: V31(V7) = θ → A126706. — : • V3118: V31(V8) = θ → A286708. — : • V3119: — : • V3150: k such that V3051(k) = 0. — : • V3151: nonprimepower k such that V3051(k) = 0. — : • V3180: Smallest ϑ(ϰ) for ω(ϰ) with n distinct prime factors. {10, 26, 64, 163, 416, ...} — : • V3181: ⌊ 1/n! × ((n+1)^(n−1) × (n+2)) ⌋, upper bound for V3180. A288784: • V3200: Necessary but insufficient condition. A288813: • V3201: Turbulent candidates in A288784. A363794: • V3202: Smallest P(n)-regular k such that θ(k) ≥ θ(P(n+1)). A289171: • V3210: “Depth-Distension” correlation for primorial(n). — : • V3220: θ(A002110(i), m) − A010846(m) for m in A288813 (Deficit of θ(m) versus θ(n, m)). A370454: • V3271: |{ k = mϰ : ω(ϰ) > 1, rad(m) | ϰ, m < p₂ }| = 1 + ⌈(log p₂)/(log p₁)⌉. A341646: V3300: ∪ { k = m × rad(k) : rad(m) | rad(k), k < rad(k)² }. A341645: V3301: ∪ { k = m × rad(k) : rad(m) | rad(k), k ≥ rad(k)² } ∪ {1}. A370266: • V3302: ∪ { k = m × rad(k) : ω(k) > 1, rad(m) | rad(k), m ≥ rad(k) } = A341645 \ A246547 = A286708 ∪ A366250. A370409: • V3303: V7 ∩ V3300. = ∪ { k = m × rad(k) : ω(k) > 1, rad(m) | rad(k), k < rad(k)² }. A366250: • V3304: ∪ { k = m × rad(k) : ω(m) ≠ ω(k) > 1, rad(m) | rad(k), k ≥ rad(k)² }. A367708: • V3305: V73 \ V3304 = A364702 \ A366250. = ∪ { k = m × rad(k) : ω(k) > 1, rad(m) | rad(k), max(p₂, q) ≤ m < rad(k) }. — : V3310: ∪ { k = m × rad(k) : rad(m) | rad(k), k ≤ rad(k)² }. A059172: V3311: ∪ { k = m × rad(k) : rad(m) | rad(k), k > rad(k)² }. A003586: V3500: R₆ = numbers m regular to 6 = (2 × 3). 3-smooth numbers. A003592: V3501: R₁₀ = numbers m regular to 10 = (2 × 5). A003591: V3502: R₁₄ = numbers m regular to 14 = (2 × 7). A003593: V3503: R₁₅ = numbers m regular to 15 = (3 × 5). A003594: V3504: R₂₁ = numbers m regular to 21 = (3 × 7). A003595: V3505: R₃₅ = numbers m regular to 35 = (5 × 7). A033845: V3550: 6R₆ = numbers m coregular to 6 = (2 × 3). A033846: V3551: 10R₁₀ = numbers m coregular to 10 = (2 × 5). A033847: V3552: 14R₁₄ = numbers m coregular to 14 = (2 × 7). A033849: V3553: 15R₁₅ = numbers m coregular to 15 = (3 × 5). A033850: V3554: 21R₂₁ = numbers m coregular to 21 = (3 × 7). A033851: V3555: 35R₃₅ = numbers m coregular to 35 = (5 × 7). A051037: V3600: R₃₀ = numbers m regular to 30 = (2 × 3 × 5). 5-smooth numbers. A108319: V3601: R₄₂ = numbers m regular to 42 = (2 × 3 × 7). A108513: V3602: R₇₀ = numbers m regular to 70 = (2 × 5 × 7). A108347: V3603: R₁₀₅ = numbers m regular to 105 = (3 × 5 × 7). A143207: V3630: 30R₃₀ = numbers m coregular to 30 = (2 × 3 × 5). A147571: V3631: 210R₂₁₀ = numbers m coregular to 210 = (2 × 3 × 5 × 7). A147572: V3632: 2310R₂₃₁₀ = numbers m coregular to 2310 = (2 × 3 × 5 × 7 × 11). — : • V3633: 30030R₃₀₀₃₀ = numbers m coregular to 30030 = (2 × 3 × 5 × 7 × 11 × 13). A002473: V3650: R₂₁₀ = numbers m regular to 210 = (2 × 3 × 5 × 7). 7-smooth numbers. A051038: V3651: R₂₃₁₀ = numbers m regular to 2310 = (2 × 3 × 5 × 7 × 11). 11-smooth numbers. A080197: V3652: R₃₀₀₃₀ = numbers m regular to 30030 = (2 × 3 × 5 × 7 × 11 × 13). 13-smooth numbers. A080681: V3653: R₅₁₀₅₁₀ = numbers m regular to 510,510 = (2 × 3 × 5 × 7 × 11 × 13 × 17). 17-smooth numbers. A080682: V3654: R₉₆₉₉₆₉₀ = numbers m regular to 9,699,690 = (2 × 3 × 5 × 7 × 11 × 13 × 17 × 19). 19-smooth numbers. A080683: V3655: R₂₂₃₀₉₂₈₇₀ = numbers m regular to 223,092,870 = (2 × 3 × 5 × 7 × 11 × 13 × 17 × 19 × 23). 23-smooth numbers. A202821: V3670: 6^n = R₆(k). A372400: • V3671: 30^n = R₃₀(k). A372401: • V3672: 210^n = R₂₁₀(k). A372402: • V3673: 2310^n = R₂₃₁₀(k). — : • V3674: 30030^n = R₃₀₀₃₀(k). A180953: V3675: 10^n = R₁₀(k). — : • V3676: 14^n = R₁₄(k). — : • V3677: 15^n = R₁₅(k). — : • V3678: 21^n = R₂₁(k). — : • V3679: 35^n = R₃₅(k). A141399: V3680: {k : k(k+1) ∈ A055932}. A374609: • V3681: {1, 2, 5, 14, 714}, squarefree terms in A141399. — : • V3682: {2, 3, 6, 15, 715}. A085152: V3683: {k : rad(k) | P(3), rad(k+1) | P(3)}. A085153: V3684: {k : rad(k) | P(4), rad(k+1) | P(4)}. A252494: V3685: {k : rad(k) | P(5), rad(k+1) | P(5)}. A252493: V3686: {k : rad(k) | P(6), rad(k+1) | P(6)}. A275156: V3687: {k : rad(k) | P(7), rad(k+1) | P(7)}. A275164: V3688: {k : rad(k) | P(8), rad(k+1) | P(8)}. — : V3689: {k : rad(k) | P(9), rad(k+1) | P(9)} = row 9 of A138180. — : V3690: {k : rad(k) | P(10), rad(k+1) | P(10)} = row 10 of A138180. — : V3691: {k : rad(k) | P(11), rad(k+1) | P(11)} = row 11 of A138180. A061987: V3692: ∆₁{k : rad(k) | P(2)}. A219794: V3693: ∆₁{k : rad(k) | P(3)}. A086247: V3694: ∆₁{k : rad(k) | P(4)}. — : • V3695: ∆₁{k : rad(k) | P(5)}. — : • V3696: ∆₁{k : rad(k) | P(6)}. — : • V3697: ∆₁{k : rad(k) | P(7)}. — : • V3698: ∆₁{k : rad(k) | P(8)}. — : • V3699: ∆₁{k : rad(k) | P(9)}. A086415: V3700: { ρ6 ↦ R6 } = richnesses of 6-regular m ∈ A3586. A352072: • V3701: { ρ12 ↦ R6 } = richnesses of 12-regular m ∈ A3586. — V3702: { ρ18 ↦ R6 } = richnesses of 18-regular m ∈ A3586. — V3703: { ρ24 ↦ R6 } = richnesses of 24-regular m ∈ A3586. — V3704: { ρ36 ↦ R6 } = richnesses of 36-regular m ∈ A3586. A117920: V3710: { ρ10 ↦ R10 } = richnesses of 10-regular m ∈ A3592. A352218: • V3711: { ρ20 ↦ R10 } = richnesses of 20-regular m ∈ A3592. — V3720: { ρ14 ↦ R14 } = richnesses of 14-regular m ∈ A3591. — V3725: { ρ15 ↦ R15 } = richnesses of 15-regular m ∈ A3593. — V3730: { ρ21 ↦ R21 } = richnesses of 21-regular m ∈ A3594. — V3735: { ρ35 ↦ R35 } = richnesses of 35-regular m ∈ A3595. — V3740: { ρ30 ↦ R30 } = richnesses of 30-regular m ∈ A051037. A352219: • V3741: { ρ60 ↦ R30 } = richnesses of 60-regular m ∈ A051037. — V3742: { ρ90 ↦ R30 } = richnesses of 90-regular m ∈ A051037. — V3743: { ρ120 ↦ R30 } = richnesses of 120-regular m ∈ A051037. — V3749: { ρ360 ↦ R30 } = richnesses of 360-regular m ∈ A051037. — V3770: { ρ210 ↦ R210 } = richnesses of 210-regular m ∈ A2473. — V3773: { ρ2520 ↦ R210 } = richnesses of 2520-regular m ∈ A2473. — V3774: { ρ5040 ↦ R210 } = richnesses of 5040-regular m ∈ A2473. — V3775: { ρ2310 ↦ R2310 } = richnesses of 2310-regular m ∈ A051038. A353383: • V3800: P12 = { k ∈ M12 : 12 ∤ k } = irregular table, positional transform of proper 12-regular numbers. A353384: • V3810: P20 = { k ∈ M20 : 20 ∤ k } = irregular table, positional transform of proper 20-regular numbers. A353385: • V3841: P60 = { k ∈ M60 : 60 ∤ k } = irregular table, positional transform of proper 60-regular numbers. — V3843: P120 = { k ∈ M120 : 120 ∤ k } = irregular table, positional transform of proper 120-regular numbers. — V3849: P360 = { k ∈ M360 : 360 ∤ k } = irregular table, positional transform of proper 360-regular numbers. — V3873: P2520 = { k ∈ M2520 : 2520 ∤ k } = irregular table, positional transform of proper 2520-regular numbers. — V3874: P5040 = { k ∈ M5040 : 5040 ∤ k } = irregular table, positional transform of proper 5040-regular numbers. A352064: V3900: Irregular triangle T(n,k) where row n lists the positions of n in A275314. (Gradus Suavitatis) A046022: V4600: zeros in A300858 (i.e., 1, 4, and the primes). A300860: • V4601: indices of records in A300858. A300861: • V4602: records in A300858. A135718: V5000: Smallest semidivisor k of n, or 0 if k does not exist. A373736: • V5001: Largest semidivisor k of n, or 0 if k does not exist. A373735: • V5002: lpf(n)^(-1 + log_lpf(n) n) = A020639(n)^A280363(n). A376422: • V5003: Numbers m whose largest semidivisor k is not a powerful. A376421: • V5004: Numbers m whose largest semidivisor k is not a prime power. — : • V5005: Smallest semidivisor k of non-prime-power n. — : • V5006: Largest semidivisor k of non-prime-power n. A359929: • V5100: Symmetric semidivisors k ⑨ A360768(n). (->V5105) A359382: • V5101: ξ₉(A360768(n)). (->V5106) — : • V5102: Symmetric Semidivisor Sum Function. — : • V5103: V5102 across V0700 (A360768). — : • V5104: Indices of records in V5104. — : • V5105: Records in V5104. — : • V5106: Symmetric Semidivisor Product Function. — : • V5107: V5106 across V0700 (A360768). — : • V5108: Indices of records in V5107. — : • V5109: Records in V5107. — • V5110: Symmetric semidivisor kernels u for squarefree ϰ ∈ V5 (i.e., A5117). — • V5111: Symmetric semidivisor kernels u for squarefree ϰ ∈ V5 (i.e., A5117). — • V5112: Symmetric semidivisor differences δ = v − u, δ | ϰ for squarefree ϰ ∈ V5 (i.e., A5117). — • V5115: Symmetric semidivisor kernel counting function. — • V5116: Symmetric semidivisor kernel counting function (across V6). — • V5119: Symmetric semidivisor kernel counting function (across V0111). A361235: • V5201: Asymmetric (Mixed) Semidivisor Counting Function ξ₇(n). A300155: • V5400: numbers with A243822(n) = A000005(n), i.e., same number of divisors and semidivisors. A299991: • V5401: numbers with A243822(n) > A000005(n), i.e., more semidivisors than divisors. A299992: • V5402: composites n with ω(n) > 1 for which A243822(n) < A000005(n), i.e., fewer semidivisors than divisors. (numbers n with ω(n) = 1, including prime n, have no semidivisors less than n). A300156: • V5403: indices of records in A299990. A300157: • V5404: records in A299990. A289280: V5410: Smallest k such that k > n and k ¦ n. A362044: • V5411: largest k such that k < m² and k ¦ m, where m = A120944(n) = V6(n). A362045: • V5412: smallest k such that k > m² and k ¦ m, where m = A120944(n) = V6(n). A362003: • V5413: Varius m such that k − m² < m, where k > m² and k ¦ m. — : • V6000: a(n) is the smallest k with ω(k) = n such that θ(k) is minimal. {1, 2, 6, 1001, 2445956099, 316850028748880382029347112807, 4256257282315965259351036612624463193990139, ...} A366786: • V6001: ϰ × lpf(ϰ) = V6(n) × V1001(V6(n)). — : • V6009: a(n) is the smallest m, product of prime k-tuples, such that θ(m) is minimal. {1, 2, 15, 1001, 46189, 1062347, 1329900201629, ...} A360480: • V6101: Symmetric Semitotative Counting Function ξ₁(n). A363844: • V6111: ξ_t(P(n)) = V61 ↦ V0111 = V243823 ↦ A2110. A360543: • V6201: Asymmetric (Mixed) Semitotative Counting Function ξ₃(n). — • V6300: Opaque semitotatives of n. { k : k ◊ n ∧ k ≤ n ∧ k ∤ (n²−1) }. A360224: • V6301: Opaque semitotative counting function ξh (n). A361080: • V6302: Highly opaque numbers. A361081: • V6303: Records in V6301. — • V6304: Opaque-semitotative dominant numbers. — • V6305: Numbers with at least as many divisors as opaque semitotatives. — • V6306: Numbers with at least as many regular numbers k ≤ n as opaque semitotatives. A294576: • V6401: odd semitotative-dominant numbers. A295221: • V6400: semitotative-parity numbers (2 × ξt (n) = n). A291989: • V6410: smallest k such that k > n and k ◊ n. A330136: • V6500: S6 = numbers m semicoprime to 6 = (2 × 3). A105115: V6501: S10 = numbers m semicoprime to 10 = (2 × 5). A316991: • V6502: S14 = numbers m semicoprime to 14 = (2 × 7). A316992: • V6503: S15 = numbers m semicoprime to 15 = (3 × 5). A306999: • V6504: S21 = numbers m semicoprime to 21 = (3 × 7). A307589: • V6505: S35 = numbers m semicoprime to 35 = (5 × 7). A330137: • V6600: S30 = numbers m semicoprime to 30 = (2 × 3 × 5). A362010: • V6601: S42 = numbers m semicoprime to 42 = (2 × 3 × 7). A362011: • V6602: S70 = numbers m semicoprime to 70 = (2 × 5 × 7). A362012: • V6603: S105 = numbers m semicoprime to 105 = (3 × 5 × 7). A084891: V6650: S210 = numbers m semicoprime to 210 = (2 × 3 × 5 × 7). — • V6651: S2310 = numbers m semicoprime to 2310 = (2 × 3 × 5 × 7 × 11). — • V6652: S30030 = numbers m semicoprime to 30,030 = (2 × 3 × 5 × 7 × 11 × 13). — • V6653: S510510 = numbers m semicoprime to 510,510 = (2 × 3 × 5 × 7 × 11 × 13 × 17). — • V6654: S9699690 = numbers m semicoprime to 9,699,690 = (2 × 3 × 5 × 7 × 11 × 13 × 17 × 19). Prime k-tuplets (continued from V067_) — : • V6700: (k=15, d = {0,2,6,8,12,18,20,26,30,32,36,42,48,50,56}). (Cf. V1530, A257304). — : • V6701: (k=15, d = {0,6,8,14,20,24,26,30,36,38,44,48,50,54,56}). (Cf. V1531, A257307). — : • V6702: (k=15, d = {0,2,6,12,14,20,24,26,30,36,42,44,50,54,56}). (Cf. V1532, A257305). — : • V6703: (k=15, d = {0,2,6,12,14,20,26,30,32,36,42,44,50,54,56}). (Cf. V1533, A257306). — : • V6704: (k=16, d = {0,4,6,10,16,18,24,28,30,34,40,46,48,54,58,60}). (Cf. V1534, A257369). — : • V6705: (k=16, d = {0,2,6,12,14,20,26,30,32,36,42,44,50,54,56,60}). (Cf. V1534, A257370). "Gauntlet" numbers. A376331: • V6750: Smallest products m of n consecutive primes p_1..p_k, where only p_1 < m^(1/n). — : • V6751: Smallest products m of n consecutive primes p_1..p_n are such that only p_n > m^(1/n). — : • V6752: A375008: • V6753: Products m of k = 3 consecutive primes p_1..p_k, where only p_1 < m^(1/k). A375975: • V6754: Products m of k = 4 consecutive primes p_1..p_k, where only p_1 < m^(1/k). A376261: • V6755: Products m of k = 5 consecutive primes p_1..p_k, where only p_1 < m^(1/k). — : • V6756: Products m of k = 6 consecutive primes p_1..p_k, where only p_1 < m^(1/k). — : • V6757: Products m of k = 7 consecutive primes p_1..p_k, where only p_1 < m^(1/k). — : • V6758: Products m of k = 8 consecutive primes p_1..p_k, where only p_1 < m^(1/k). — : • V6759: Products m of k = 9 consecutive primes p_1..p_k, where only p_1 < m^(1/k). A365783: • V7000: V0220 ↦ V7 = squarefree kernel of tantus numbers. (->V0705->V0710) A365784: • V7001: Kernel ratio of tantus numbers V7(n)/V7000(n). (->V0706->V0711) A365710: • V7002: Second smallest prime factor of tantus k. A360769: • V7003: Odd tantus numbers. (->V0703->V7005) A363101: • V7004: Even tantus numbers. (->V0702) A367455: • V7005: { k : Ω(k) = ω(k) > 1, p₂ > q } = { k : 6 ∤ k, Ω(k) = ω(k) > 1 }. (->V0716->V7008->V7007) — : • V7006: { k : Ω(k) = ω(k) > 1, p₂ < q } = { k = m × 6χ : Ω(χ) = ω(χ), rad(m) | 6χ }. (->V0717) A369954: • V7007: { k : Ω(k) = ω(k) > 1, (k,6) = 1 } (->V7008) A376720: • V7008: Kernel product of tantus numbers V7(n) × V7000(n) ⊂ V8. (in A372404) A370395: • V7009: Tantus sandwiched between twin primes. A365790: • V7010: V31 ↦ V7 = regular counting function of tantus numbers. (->V0708->V0713) A365791: • V7011: V34 ↦ V7 = coregular counting function of tantus numbers. (->V0709->V0714) — : • V7012: Odd thick tantus numbers. (->V0705->V7009) A361487: • V7014: Odd strong tantus numbers. (->V0704->V7007) A365785: • V7016: V6(k) = V7000(n). (->V0707->V0712->V7003) A362432: • V7019: Smallest k < A126706(n) such that rad(k) = rad(A126706(n)) and k mod n != 0 A358089: • V7020: First differences of tantus numbers V7. (->V0079->V7012) A369516: • V7021: Tantus k such that neither (k-1) nor (k+1) are tantus. A369276: • V7022: Tantus k in A126706 such that (k-1) or (k+1) are tantus. A370372: • V7023: Run lengths of 1s in V7020. Row lengths of A369276. A356322: • V7024: Smallest number that starts a run of exactly n consecutive tantus numbers. (->V0078->V7013) — : • V7071: Odd terms in V0770. (->V0771) A365786: • V8000: V0220 ↦ V8 = squarefree kernel of plenus numbers. (->V0850->V0820) A365787: • V8001: V8(n)/V0850(n). (->V0855->V0825) — : • V8002: Second smallest prime factor of plenus k. — : • V8003: V6(k) = V0850(n). (->V0856->V0826) A363216: • V8004: Even plenus numbers. (->V82) A363217: • V8005: Odd plenus numbers. (->V83) A365789: • V8006: k : V0109(k) = V0850(n). (->V0857->V0827) — : • V8007: — : • V8008: A113839: • V8009: Plenus sandwiched between twin primes. (A113839\{4}) A365792: • V8010: V31 ↦ V8 = regular counting function of plenus numbers. (->V0858->V0828) A365793: • V8011: V34 ↦ V8 = coregular counting function of plenus numbers. (->V0859->V0829) A358173: • V8012: First differences of plenus numbers V8. (->V0089) A374609: • V6___: Smaller oblongizing factor of oblong primorials. I dedicate the Vinci Constitutive Catalog to the memory of Mary Katherine De Vlieger (Nuzzo), elder sister to my father, who suggested that composite numbers were more complex to think about than primes, which in Easter 1982 ultimately inspired this work. Updated 202409200830