A277239 -id:A277239 - cp's OEIS frontend

A002774 Number of bipartite partitions of n white objects and n black ones.

1, 2, 9, 31, 109, 339, 1043, 2998, 8406, 22652, 59521, 151958, 379693, 927622, 2224235, 5236586, 12130780, 27669593, 62229990, 138095696, 302673029, 655627975, 1404599867, 2977831389, 6251060785, 12999299705, 26791990052, 54750235190, 110977389012

Offset: 0

N. J. A. Sloane

nonn

Number of ways to factor p^n * q^n where p and q are distinct primes.

M. S. Cheema and H. Gupta, Tables of Partitions of Gaussian Integers. National Institute of Sciences of India, Mathematical Tables, Vol. 1, New Delhi, 1956, p. 1.
D. E. Knuth, The Art of Computer Programming, Vol. 4A, Table A-1, see p(n,n), page 778. - N. J. A. Sloane, Dec 30 2018
A. Murthy, Generalization of partition function, introducing Smarandache factor partitions. Smarandache Notions Journal, Vol. 11, No. 1-2-3, Spring 2000.
A. Murthy, Program for finding out the number of Smarandache factor partitions. (To be published in Smarandache Notions Journal).
Amarnath Murthy and Charles Ashbacher, Generalized Partitions and Some New Ideas on Number Theory and Smarandache Sequences, Hexis, Phoenix; USA 2005. See Section 1.4, 1.14.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..400 (terms 0..100 from Alois P. Heinz)
F. C. Auluck, On partitions of bipartite numbers, Proc. Cambridge Philos. Soc. 49, (1953). 72-83.
F. C. Auluck, On partitions of bipartite numbers, annotated scan of a few pages.
F. C. Auluck, On partitions of bipartite numbers, Mathematical Proceedings of the Cambridge Philosophical Society, Volume 49, Issue 01, January 1953, pp. 72-83. (full article)
Katherine Ormeño Bastías, Paul Martin, and Steen Ryom-Hansen, On the spherical partition algebra, arXiv:2402.01890 [math.RT], 2024.
M. S. Cheema and H. Gupta, Tables of Partitions of Gaussian Integers. National Institute of Sciences of India, Mathematical Tables, Vol. 1, New Delhi, 1956 (Annotated scanned pages from, plus a review)
Edward Pearce-Crump, Permutation Equivariant Neural Networks for Symmetric Tensors, arXiv:2503.11276 [cs.LG], 2025. See p. 20.

Cf. A005380.
Cf. A219554. Column k=2 of A219727. - Alois P. Heinz, Nov 26 2012
Cf. also A000041, A000070, A000291, A000412, A000465, A000491, A002755, A002756, A002757, A002758, A002759, A277239.
Main diagonal of A054225 if that entry is drawn as a square array. - N. J. A. Sloane, Dec 30 2018

with(numtheory):
b:= proc(n, k) option remember; `if`(n>k, 0, 1) +`if`(isprime(n), 0,
      add(`if`(d>k, 0, b(n/d, d)), d=divisors(n) minus {1, n}))
    end:
a:= n-> b(6^n$2):
seq(a(n), n=0..30);  # Alois P. Heinz, Jun 27 2013

max = 26; se = Series[ Sum[ Log[1 - x^(n-k)*y^k], {n, 1, 2max}, {k, 0, n}], {x, 0, 2max}, {y, 0, 2max}]; coes = CoefficientList[ Series[ Exp[-se], {x, 0, 2max}, {y, 0, 2max}], {x, y}]; a[n_] := coes[[n+1, n+1]]; Table[a[n], {n, 0, max} ](* Jean-François Alcover, Dec 06 2011 *)

a(n) = A054225(2n, n) = A091437(2n).
a(n) ~ Zeta(3)^(19/36) * exp(3*Zeta(3)^(1/3) * n^(2/3) + Pi^2 * n^(1/3) / (6*Zeta(3)^(1/3)) + Zeta'(-1) - Pi^4/(432*Zeta(3))) / (sqrt(3) * (2*Pi)^(3/2) * n^(55/36)). - Vaclav Kotesovec, Jan 30 2016
Formula (25) in the article by Auluck is incorrect. The correct formula is: p(n,n) ~ c^(19/12) * exp(3*c*n^(2/3) + 3*d*n^(1/3) + Zeta'(-1) - 3*d^2/(4*c)) / (sqrt(3) * (2*Pi)^(3/2) * n^(55/36)), where c = Zeta(3)^(1/3), d = Zeta(2)/(3*c). Also formula (24) is incorrect. - Vaclav Kotesovec, Jan 30 2016
From Vaclav Kotesovec, Feb 04 2016: (Start)
The correct formula (24) is p(m,n) ~ c^(7/4)/(2*Pi*sqrt(3)) * exp(3*c*(m*n)^(1/3) + 3*d*(m+n)/(2*(m*n)^(1/3)) - 19*log(m*n)/24 - ((m/n - 2*n/m)*log(m) + (n/m - 2*m/n)*log(n))/36 - (m/n + n/m)*(log(c)/12 + Zeta'(-1) - 1/12 + 3*d^2/(4*c)) + 3*d^2/(4*c) - 3*log(2*Pi)/4 + fi((n/m)^(1/2))),
where m and n are of the same order, c = Zeta(3)^(1/3), d = Zeta(2)/(3*c) and fi(alfa) = Integral_{t=0..infinity} (1/t)*(1/(exp(alfa*t)-1)/(exp(t/alfa)-1) - (alfa/t)/(exp(alfa*t)-1) - ((1/alfa)/t)/(exp(t/alfa)-1) + 1/t^2 + (1/4)/(exp(alfa*t)-1) + (1/4)/(exp(t/alfa)-1) - (alfa/4)/t - ((1/4)/alfa)/t).
If m = n then alfa = 1 and fi(1) = 3*Zeta'(-1) + log(2*Pi)/4 - 1/6.
For the asymptotic formula for fixed m see A054225.
(End)

Corrected using A000491.

Edited by Christian G. Bower, Jan 08 2004

A101427 Number of different cuboids with volume (pq)^n, where p,q are distinct prime numbers.

Original entry on oeis.org

1, 2, 8, 19, 42, 78, 139, 224, 350, 517, 744, 1032, 1405, 1862, 2432, 3115, 3942, 4914, 6067, 7400, 8954, 10729, 12768, 15072, 17689, 20618, 23912, 27571, 31650, 36150, 41131, 46592, 52598, 59149, 66312, 74088, 82549, 91694, 101600, 112267, 123774

Offset: 0

Anthony C Robin, Jan 17 2005

nonn

Subsequence of A034836, which gives the number of cuboids for volume n.

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Geoffrey B. Campbell, Vector Partition Identities for 2D, 3D and nD Lattices, arXiv:2302.01091 [math.CO], 2023.
Index entries for linear recurrences with constant coefficients, signature (2,1,-3,-1,1,3,-1,-2,1).

Cf. A034836, A101423, A101424, A101425, A101426.

Column k=3 of A277239.

a[n_] := Switch[Mod[n, 6], 0, n+1, 1|5, 3n/4 + 7/24, 2|4, n+2/3, 3, 3n/4 + 5/8] + n^4/24 + n^3/4 + 2n^2/3; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Oct 06 2016, after Frederic Solbes' formula *)

a(n) = if (n % 3, ((n+2)^2*(n+1)^2 + 12*(n\2+1)^2)/24, ((n+2)^2*(n+1)^2 + 12*(n\2+1)^2+8)/24); \\ Michel Marcus, Mar 18 2014

If n is a multiple of 3, a(n) = ((n+2)^2*(n+1)^2 + 12*(floor(n/2)+1)^2+8)/24, otherwise a(n) = ((n+2)^2*(n+1)^2 + 12*(floor(n/2)+1)^2)/24. - Frederic Solbes, Mar 18 2014
G.f.: -(x^6+3*x^4+4*x^3+3*x^2+1)/((x^2+x+1)*(x+1)^2*(x-1)^5). - Colin Barker, Mar 27 2014
From Daniel Mondot, Sep 20 2016: (Start)
a(n) = a(n-1) + 2*a(n-2) - a(n-3) - 2*a(n-4) - a(n-5) + 2*a(n-6) + a(n-7) - a(n-8) + 12, n>=8.
a(n) = 4*a(n-6) - 6*a(n-12) + 4*a(n-18) - a(n-24) + 1296, n>=24. (End)

Extended by Ray Chandler, Dec 17 2008
Edited by Ray Chandler, Dec 19 2008
a(0) = 1 prepended by Daniel Mondot, Sep 20 2016

A254686 Number of ways to put n red and n blue balls into n indistinguishable boxes.

Original entry on oeis.org

1, 1, 5, 19, 74, 248, 814, 2457, 7168, 19928, 53688, 139820, 354987, 878434, 2128102, 5052010, 11781881, 27019758, 61035671, 135928105, 298784144, 648726349, 1392474574, 2956730910, 6214668074, 12937060340, 26686392239, 54572423946, 110680119454, 222710856175, 444776676764

Offset: 0

Brian Chen, Feb 08 2015

nonn

See a comment on A254811 about multiset partitions and the Knuth reference. - Wolfdieter Lang, Mar 26 2015

			For n = 2 the a(2) = 5 ways to put 2 red balls and 2 blue balls into 2 indistinguishable boxes are (RRBB)(), (RRB)(B), (RBB)(R), (RR)(BB), (RB)(RB).

Brian Chen, Table of n, a(n) for n = 0..64

Cf. A002774, A060150, A254811.
Column k=2 of A256384.
Main diagonal of A277239.

with(numtheory):
b:= proc(n, k, i) option remember;
      `if`(n>k, 0, 1) +`if`(isprime(n) or i<2, 0, add(
      `if`(d>k, 0, b(n/d, d, i-1)), d=divisors(n) minus {1, n}))
    end:
a:= n-> b(6^n$2,n):
seq(a(n), n=0..20);  # Alois P. Heinz, Mar 26 2015

b[n_, k_, i_] := b[n, k, i] = If[n > k, 0, 1] + If[PrimeQ[n] || i < 2, 0, Sum[If[d > k, 0, b[n/d, d, i - 1]], {d, Divisors[n] [[2 ;; -2]]}]]; a[n_] := b[6^n, 6^n, n]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jan 08 2016, after Alois P. Heinz *)

A277240 Number of factorizations of m^n into exactly four factors, where m is a product of two distinct primes.

Original entry on oeis.org

1, 2, 9, 27, 74, 168, 363, 703, 1297, 2247, 3742, 5967, 9241, 13859, 20307, 29052, 40786, 56187, 76233, 101858, 134377, 175068, 225640, 287772, 363673, 455482, 565977, 697875, 854594, 1039500, 1256787, 1510547, 1805833, 2147607, 2541870, 2994543, 3512737

Offset: 0

Alois P. Heinz, Oct 06 2016

nonn

			a(2) = 9: (2*3)^2 = 2*2*3*3 = 1*3*3*4 = 1*2*3*6 = 1*2*2*9 = 1*1*4*9 = 1*1*6*6 = 1*1*2*18 = 1*1*3*12 = 1*1*1*36.

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (2,1,-2,-2,-2,5,2,0,-2,-5,2,2,2,-1,-2,1)

Column k=4 of A277239.

LinearRecurrence[{2,1,-2,-2,-2,5,2,0,-2,-5,2,2,2,-1,-2,1},{1,2,9,27,74,168,363,703,1297,2247,3742,5967,9241,13859,20307,29052},40] (* Harvey P. Dale, May 21 2024 *)

G.f.: -(x^12 +4*x^10 +9*x^9 +17*x^8 +17*x^7 +24*x^6 +17*x^5 +17*x^4 +9*x^3 +4*x^2 +1) / ((x^2+1) *(x^2+x+1)^2 *(x+1)^3 *(x-1)^7).

A277241 Number of factorizations of m^n into exactly five factors, where m is a product of two distinct primes.

Original entry on oeis.org

1, 2, 9, 30, 95, 248, 614, 1367, 2879, 5674, 10681, 19169, 33203, 55478, 90067, 142224, 219388, 330943, 489716, 711518, 1017241, 1432328, 1989400, 2727644, 3696314, 4953733, 6571851, 8635136, 11245971, 14523411, 18610003, 23669939, 29897469, 37514880, 46782198

Offset: 0

Alois P. Heinz, Oct 06 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Column k=5 of A277239.

A277242 Number of factorizations of m^n into exactly six factors, where m is a product of two distinct primes.

Original entry on oeis.org

1, 2, 9, 31, 105, 300, 814, 1996, 4642, 10105, 20975, 41455, 78849, 144371, 256079, 440704, 738772, 1208149, 1932864, 3029214, 4660063, 7045203, 10483231, 15368227, 22222681, 31722589, 44745412, 62407716, 86132422, 117703074, 159357465, 213863183, 284644763

Offset: 0

Alois P. Heinz, Oct 06 2016

nonn

Column k=6 of A277239.

A277243 Number of factorizations of m^n into exactly seven factors, where m is a product of two distinct primes.

Original entry on oeis.org

1, 2, 9, 31, 108, 325, 938, 2457, 6128, 14358, 32123, 68488, 140478, 277278, 529624, 980476, 1765492, 3097071, 5306548, 8893500, 14607217, 23542097, 37285849, 58094004, 89148980, 134862913, 201312513, 296746270, 432288197, 622756805, 887772827, 1253040268

Offset: 0

Alois P. Heinz, Oct 06 2016

nonn

Column k=7 of A277239.

A277244 Number of factorizations of m^n into exactly eight factors, where m is a product of two distinct primes.

Original entry on oeis.org

1, 2, 9, 31, 109, 335, 1002, 2741, 7168, 17685, 41801, 94377, 205277, 430087, 872302, 1714941, 3278477, 6103521, 11091375, 19700943, 34265319, 58429093, 97816791, 160944051, 260558245, 415434960, 652940560, 1012418264, 1549884617, 2344153951, 3505141354

Offset: 0

Alois P. Heinz, Oct 06 2016

nonn

Column k=8 of A277239.

A277245 Number of factorizations of m^n into exactly nine factors, where m is a product of two distinct primes.

Original entry on oeis.org

1, 2, 9, 31, 109, 338, 1029, 2889, 7797, 19928, 48965, 115275, 262034, 574803, 1222170, 2521350, 5060938, 9896451, 18891871, 35249977, 64390786, 115284416, 202560300, 349637130, 593499173, 991625986, 1632248998, 2648936320, 4241607523, 6705867270, 10474314965

Offset: 0

Alois P. Heinz, Oct 06 2016

nonn

Column k=9 of A277239.

A277246 Number of factorizations of m^n into exactly ten factors, where m is a product of two distinct primes.

Original entry on oeis.org

1, 2, 9, 31, 109, 339, 1039, 2957, 8135, 21280, 53688, 130174, 305539, 693598, 1528901, 3274785, 6832548, 13900570, 27626151, 53692474, 102193925, 190678199, 349172204, 628122125, 1111050987, 1934048642, 3315905471, 5603441375, 9339723645, 15364504830

Offset: 0

Alois P. Heinz, Oct 06 2016

nonn

Column k=10 of A277239.

cp's OEIS Frontend

A002774 Number of bipartite partitions of n white objects and n black ones.

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A101427 Number of different cuboids with volume (pq)^n, where p,q are distinct prime numbers.

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A254686 Number of ways to put n red and n blue balls into n indistinguishable boxes.

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A277240 Number of factorizations of m^n into exactly four factors, where m is a product of two distinct primes.

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A277241 Number of factorizations of m^n into exactly five factors, where m is a product of two distinct primes.

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A277242 Number of factorizations of m^n into exactly six factors, where m is a product of two distinct primes.

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A277243 Number of factorizations of m^n into exactly seven factors, where m is a product of two distinct primes.

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A277244 Number of factorizations of m^n into exactly eight factors, where m is a product of two distinct primes.

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A277245 Number of factorizations of m^n into exactly nine factors, where m is a product of two distinct primes.

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A277246 Number of factorizations of m^n into exactly ten factors, where m is a product of two distinct primes.

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