A007426 -id:A007426 - cp's OEIS frontend

A175596 Partial products of A007425.

1, 3, 9, 54, 162, 1458, 4374, 43740, 262440, 2361960, 7085880, 127545840, 382637520, 3443737680, 30993639120, 464904586800, 1394713760400, 25104847687200, 75314543061600, 1355661775108800, 12200955975979200, 109808603783812800, 329425811351438400, 9882774340543152000, 59296646043258912000, 533669814389330208000, 5336698143893302080000, 96060566590079437440000, 288181699770238312320000, 7780905893796434432640000

Offset: 1

Jonathan Vos Post, Dec 03 2010

nonn

Partial products of the number of ordered factorizations of n as a product of 3 terms.

a(n) is also the determinant of the symmetric n X n matrix M defined by M(i,j) = d_4(gcd(i,j)) for 1 <= i,j <= n, where d_4(n) = A007426(n). - Enrique Pérez Herrero, Jan 20 2013

			a(8) = 1 * 3 * 3 * 6 * 3 * 9 * 3 * 10 = 43740 = 2^2 * 3^7 * 5.

Antal Bege, Hadamard product of GCD matrices, Acta Univ. Sapientiae, Mathematica, 1, 1 (2009) 43-49.

Cf. A000005, A007425, A007426, A061201 (partial sums), A127270, A143354.

Cf. A066843.

Table[Product[Sum[DivisorSigma[0, d], {d, Divisors[k]}], {k, 1, n}], {n, 1, 30}] (* Vaclav Kotesovec, Sep 03 2018 *)

f(n) = sumdiv(n, k, numdiv(k)); \\ A007425
a(n) = prod(k=1, n, f(k)); \\ Michel Marcus, Mar 23 2021

a(n) = Product_{i=1..n} A007425(i).

a(n) = Product_{prime p<=n} Product_{k=1..floor(log_p(n))} (1 + 2/k)^floor(n/p^k). - Ridouane Oudra, Mar 23 2021

A304964 Expansion of 1/(1 - Sum_{i>=1, j>=1, k>=1, l>=1} x^(ijk*l)).

Original entry on oeis.org

1, 1, 5, 13, 47, 133, 443, 1333, 4263, 13143, 41419, 128791, 403815, 1259639, 3941579, 12310299, 38492034, 120271953, 375964616, 1174935195, 3672413322, 11477465221, 35872928244, 112117013835, 350417746650, 1095202995267, 3422999582632, 10698350241417, 33437065631262, 104505382585023

Offset: 0

Ilya Gutkovskiy, May 22 2018

nonn

Invert transform of A007426.

N. J. A. Sloane, Transforms
Index entries for sequences related to compositions

Cf. A000005, A007426, A011782, A129921, A280486, A280487, A304963.

A:= proc(n, k) option remember; `if`(k=1, 1,
      add(A(d, k-1), d=numtheory[divisors](n)))
    end:
a:= proc(n) option remember; `if`(n=0, 1,
      add(A(j, 4)*a(n-j), j=1..n))
    end:
seq(a(n), n=0..35);  # Alois P. Heinz, May 22 2018

nmax = 29; CoefficientList[Series[1/(1 - Sum[x^(i j k l), {i, 1, nmax}, {j, 1, nmax/i}, {k, 1, nmax/i/j}, {l, 1, nmax/i/j/k}]), {x, 0, nmax}], x]
nmax = 29; CoefficientList[Series[1/(1 - Sum[Sum[DivisorSigma[0, d] DivisorSigma[0, k/d], {d, Divisors[k]}] x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[Sum[DivisorSigma[0, d] DivisorSigma[0, k/d], {d, Divisors[k]}] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 29}]

G.f.: 1/(1 - Sum_{k>=1} A007426(k)*x^k).

A321240 Expansion of Product_{i>=1, j>=1, k>=1, l>=1} (1 + x^(ijkl))/(1 - x^(ijkl)).

Original entry on oeis.org

1, 2, 10, 26, 86, 210, 594, 1394, 3530, 8006, 18842, 41258, 92190, 195714, 419538, 867050, 1797568, 3625758, 7311382, 14431294, 28416514, 55010142, 106101558, 201814518, 382213566, 715473554, 1333083950, 2459265058, 4515151234, 8218572030, 14888270366, 26766878302

Offset: 0

Seiichi Manyama, Nov 01 2018

nonn

Convolution of the sequences A280486 and A280487.

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A007426, A015128, A280486, A280487, A301554, A305050.

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(   (&*[(&*[(&*[(&*[(1+x^(i*j*k*l))/(1-x^(i*j*k*l)): i in [1..m]]): j in [1..m]]): k in [1..m]]): l in [1..m]]))); // G. C. Greubel, Nov 01 2018

With[{nmax=50}, CoefficientList[Series[Product[(1 + x^(i*j*k*l))/(1 - x^(i*j*k*l)), {i,1,nmax}, {j,1,nmax/i}, {k,1,nmax/i/j}, {l,1,nmax/i/j/k}], {x,0,nmax}], x]] (* G. C. Greubel, Nov 01 2018 *)

m=50; x='x+O('x^m); Vec(prod(k=1,m, ((1+x^k)/(1-x^k))^ sumdiv(k, d, numdiv(k/d)*numdiv(d)))) \\ G. C. Greubel, Nov 01 2018

G.f.: Product_{k>=1} ((1 + x^k)/(1 - x^k))^A007426(k).

A328486 Dirichlet g.f.: zeta(s)^4 * (1 - 2^(-s))^2.

Original entry on oeis.org

1, 2, 4, 3, 4, 8, 4, 4, 10, 8, 4, 12, 4, 8, 16, 5, 4, 20, 4, 12, 16, 8, 4, 16, 10, 8, 20, 12, 4, 32, 4, 6, 16, 8, 16, 30, 4, 8, 16, 16, 4, 32, 4, 12, 40, 8, 4, 20, 10, 20, 16, 12, 4, 40, 16, 16, 16, 8, 4, 48, 4, 8, 40, 7, 16, 32, 4, 12, 16, 32, 4, 40, 4, 8, 40, 12, 16, 32, 4, 20

Offset: 1

Ilya Gutkovskiy, Oct 16 2019

Dirichlet convolution of A001227 with itself.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A001227, A007426, A070288, A318366, A328487.

with(numtheory):
b:= proc(n) option remember; tau(2*n)-tau(n) end:
a:= n-> add(b(d)*b(n/d), d=divisors(n)):
seq(a(n), n=1..100);  # Alois P. Heinz, Oct 16 2019

nmax = 80; A001227 = Table[DivisorSum[n, Mod[#, 2] &], {n, 1, nmax}]; Table[DivisorSum[n, A001227[[#]] A001227[[n/#]] &], {n, 1, nmax}]
f[2, e_] := e + 1; f[p_, e_] := (e + 1)*(e + 2)*(e + 3)/6; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Nov 30 2020 *)

a(n) = Sum_{d|n} A001227(d) * A001227(n/d).
Sum_{k=1..n} a(k) ~ n * (log(n)^3/24 + (g/2 + log(2)/4 - 1/8)* log(n)^2 + (1/4 - g + 3*g^2/2 - log(2)/2 + 2*g*log(2) - sg1)* log(n) - 1/4 + (1 - 2*log(2))*g + (3*log(2) - 3/2)*g^2 + g^3 + log(2)/2 - log(2)^3/6 + (1 - 3*g - 2*log(2))* sg1 + sg2/2), where g is the Euler-Mascheroni constant A001620 and sg1, sg2 are the Stieltjes constants, see A082633 and A086279. - Vaclav Kotesovec, Oct 17 2019
Multiplicative with a(2^e) = e + 1, and a(p^e) = (e + 1)*(e + 2)*(e + 3)/6 for odd primes p. - Amiram Eldar, Nov 30 2020

A334115 Numbers that can be written as a product of tetrahedral numbers.

Original entry on oeis.org

0, 1, 4, 10, 16, 20, 35, 40, 56, 64, 80, 84, 100, 120, 140, 160, 165, 200, 220, 224, 256, 286, 320, 336, 350, 364, 400, 455, 480, 560, 640, 660, 680, 700, 800, 816, 840, 880, 896, 969, 1000, 1024, 1120, 1140, 1144, 1200, 1225, 1280, 1330, 1344, 1400, 1456, 1540, 1600

Offset: 1

Ilya Gutkovskiy, Apr 14 2020

nonn

For n > 1, numbers that appear at least once in A007426.

David A. Corneth, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Tetrahedral Number

Cf. A000292, A007426, A085782, A265010.

More terms from David A. Corneth, Mar 22 2021

A383658 Denominator of Dirichlet g.f.: Sum_{n>=1} a(n)/n^s = zeta(s)^(3/2).

Original entry on oeis.org

1, 2, 2, 8, 2, 4, 2, 16, 8, 4, 2, 16, 2, 4, 4, 128, 2, 16, 2, 16, 4, 4, 2, 32, 8, 4, 16, 16, 2, 8, 2, 256, 4, 4, 4, 64, 2, 4, 4, 32, 2, 8, 2, 16, 16, 4, 2, 256, 8, 16, 4, 16, 2, 32, 4, 32, 4, 4, 2, 32, 2, 4, 16, 1024, 4, 8, 2, 16, 4, 8, 2, 128, 2, 4, 16, 16, 4

Offset: 1

Vaclav Kotesovec, May 04 2025

Is this a duplicate of A046644 (the first 8192 entries are the same)? - R. J. Mathar, May 06 2025

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Cf. A383657.
Cf. A046643, A046644, A256688, A256689, A256690, A256691, A256692, A256693.
Cf. A000005, A007425, A007426, A061200, A034695.

coeff=CoefficientList[Series[1/(1-x)^(3/2),{x,0,20}]//Normal,x]; dptTerm[n_]:=Module[{flist=FactorInteger[n]},If[n==1,coeff[[1]],Denominator[Times@@(coeff[[flist[[All,2]]+1]])]]];Array[dptTerm,77] (* Shenghui Yang, May 04 2025 *)

for(n=1, 100, print1(denominator(direuler(p=2, n, 1/(1-X)^(3/2))[n]), ", "))

Sum_{k=1..n} A383657(k)/A383658(k) ~ 2*n*sqrt(log(n)/Pi) * (1 - (1 - 3*gamma/2)/(2*log(n))), where gamma is the Euler-Mascheroni constant A001620.

A098052 T(n,k) counts the solid partitions of n that can be extended to a solid partition of n+1 in exactly (k+3) ways. Equivalently, the number of solid partitions of n that have exactly k+3 partitions of n+1 majoring them.

Original entry on oeis.org

1, 4, 4, 6, 10, 12, 0, 4, 4, 30, 12, 12, 0, 0, 1, 16, 48, 18, 48, 0, 6, 4, 4, 70, 72, 100, 27, 12, 22, 20, 102, 114, 232, 76, 66, 68, 6, 10, 114, 231, 448, 232, 180, 201, 48, 16, 204, 330, 728, 628, 462, 546, 184, 24

Offset: 4

Wouter Meeussen, Sep 11 2004

Row sums are A000293 (solid partitions) by definition.
First column is conjectured to be A007426 = tau_4(n).
All solid partitions can be extended in at least 4 ways (hence the offset 4).

			T(5,7)=1 because there is only 1 solid partition of 5 [{{2, 1}, {1}}, {{1}}] that can be extended to a solid partition of 6 in exactly (7+3 =10) ways:
  [{{2,1},{2}},{{1}}], [{{2,1},{1,1}},{{1}}], [{{2,2},{1}},{{1}}],
  [{{3,1},{1}},{{1}}], [{{2,1,1},{1}},{{1}}], [{{2,1},{1},{1}},{{1}}],
  [{{2,1},{1}},{{2}}], [{{2,1},{1}},{{1,1}}], [{{2,1},{1}},{{1},{1}}],
  [{{2,1},{1}},{{1}},{{1}}].
Table starts
  1;
  4;
  4,6;
  10,12,0,4;
  4,30,12,12,0,0,1;
  16,48,18,48,0,6,4;
  4,70,72,100,27,12,22;
  20,102,114,232,76,66,68,6;
  ...

Wouter Meeussen, Mma functions for plane and solid partitions

Cf. A000029, A007426, A097994.

(* functions 'solidform' and 'coversplaneQ', see A096574 *)
Table[ Rest@BinCounts[Count[Flatten[solidformBTK/@IntegerPartitions[n+1]],q_/;coverssolidQ[q,#]]&/@Flatten[solidformBTK/@IntegerPartitions[n]]] ,{n,1,8}] (* Wouter Meeussen, Feb 03 2025 *)

A318966 Expansion of e.g.f. Product_{i>=1, j>=1, k>=1} 1/(1 - x^(ijk))^(1/(ijk)).

Original entry on oeis.org

1, 1, 5, 21, 165, 1077, 11457, 103905, 1345257, 15834825, 237535389, 3372509709, 59235634125, 979573962429, 19224990899865, 366788042231193, 8019002662543953, 171360055378885905, 4132946756763614133, 97947895990285022085, 2576516749059849502581, 67124117357620005459141

Offset: 0

Ilya Gutkovskiy, Sep 06 2018

nonn

Lida Ahmadi, Ricardo Gómez Aíza, and Mark Daniel Ward, A unified treatment of families of partition functions, La Matematica (2024). Preprint available as arXiv:2303.02240 [math.CO], 2023.

Cf. A000005, A007425, A007426, A028342, A174465, A318413, A318695, A318967.

a:=series(mul(mul(mul(1/(1-x^(i*j*k))^(1/(i*j*k)),k=1..21),j=1..50),i=1..50),x=0,22): seq(n!*coeff(a,x,n),n=0..21); # Paolo P. Lava, Apr 02 2019

nmax = 21; CoefficientList[Series[Product[Product[Product[1/(1 - x^(i j k))^(1/(i j k)), {i, 1, nmax}], {j, 1, nmax}], {k, 1, nmax} ], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 21; CoefficientList[Series[Product[1/(1 - x^k)^(Sum[DivisorSigma[0, d], {d, Divisors[k]}]/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = (n - 1)! Sum[Sum[Sum[DivisorSigma[0, j], {j, Divisors[d]}], {d, Divisors[k]}] a[n - k]/(n - k)!, {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 21}]

E.g.f.: Product_{k>=1} 1/(1 - x^k)^(tau_3(k)/k), where tau_3 = A007425.

E.g.f.: exp(Sum_{k>=1} ( Sum_{d|k} Sum_{j|d} tau(j) ) * x^k/k), where tau = number of divisors (A000005).

A321302 Expansion of Product_{i>=1, j>=1, k>=1, l>=1} (1 - x^(ijkl))/(1 + x^(ijkl)).

Original entry on oeis.org

1, -2, -6, 6, 14, 30, -14, -98, -86, -150, 282, 486, 502, 670, -1118, -1226, -4396, -3814, 1326, 3834, 20354, 16330, 18334, -6606, -45658, -60762, -121770, -60122, -22750, 160314, 303638, 435450, 542336, 162782, -45830, -1090994, -1576378, -2608146, -2408142, -988202, 479834

Offset: 0

Seiichi Manyama, Nov 03 2018

sign

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Convolution inverse of A321240.

Cf. A007426, A320908, A321241.

\\ here b(n) is A007426.
b(n)={vecprod(apply(e->binomial(e+3, 3), factor(n)[,2]))}
seq(n)={Vec(prod(k=1, n, ((1 - x^k)/(1 + x^k) + O(x*x^n))^b(k)))} \\ Andrew Howroyd, Nov 06 2018

G.f.: Product_{k>=1} ((1 - x^k)/(1 + x^k))^A007426(k).

A326417 Dirichlet g.f.: zeta(s)^4 * (1 - 2^(-s)).

Original entry on oeis.org

1, 3, 4, 6, 4, 12, 4, 10, 10, 12, 4, 24, 4, 12, 16, 15, 4, 30, 4, 24, 16, 12, 4, 40, 10, 12, 20, 24, 4, 48, 4, 21, 16, 12, 16, 60, 4, 12, 16, 40, 4, 48, 4, 24, 40, 12, 4, 60, 10, 30, 16, 24, 4, 60, 16, 40, 16, 12, 4, 96, 4, 12, 40, 28, 16, 48, 4, 24, 16, 48, 4, 100, 4, 12, 40

Offset: 1

Ilya Gutkovskiy, Oct 18 2019

Inverse Moebius transform applied twice to A001227.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A001227, A007425, A007426, A133700, A280486, A318845.

Table[Sum[DivisorSigma[0, n/d] Total[Mod[Divisors[d], 2]], {d, Divisors[n]}], {n, 1, 75}]
nmax = 75; A007425 = Table[DivisorSum[n, DivisorSigma[0, #] &], {n, 1, nmax}]; Table[DivisorSum[n, A007425[[#]] &, OddQ[n/#] &], {n, 1, nmax}]
f[2, e_] := (e + 1)*(e + 2)/2; f[p_, e_] := (e + 1)*(e + 2)*(e + 3)/6; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Dec 02 2020 *)

G.f.: Sum_{k>=1} tau_3(k) * x^k / (1 - x^(2*k)), where tau_3 = A007425.
a(n) = tau_4(n) if n odd, tau_4(n) - tau_4(n/2) if n even, where tau_4 = A007426.
a(n) = Sum_{d|n, n/d odd} tau_3(d).
a(n) = Sum_{d|n} A000005(n/d) * A001227(d).
Product_{n>=1} 1 / (1 - x^n)^a(n) = g.f. for A280486.
Multiplicative with a(2^e) = (e+1)*(e+2)/2, and a(p^e) = (e+1)*(e+2)*(e+3)/6 for odd primes p. - Amiram Eldar, Dec 02 2020

cp's OEIS Frontend

A175596 Partial products of A007425.

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A304964 Expansion of 1/(1 - Sum_{i>=1, j>=1, k>=1, l>=1} x^(i*j*k*l)).

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A321240 Expansion of Product_{i>=1, j>=1, k>=1, l>=1} (1 + x^(i*j*k*l))/(1 - x^(i*j*k*l)).

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Magma

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A328486 Dirichlet g.f.: zeta(s)^4 * (1 - 2^(-s))^2.

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A334115 Numbers that can be written as a product of tetrahedral numbers.

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A383658 Denominator of Dirichlet g.f.: Sum_{n>=1} a(n)/n^s = zeta(s)^(3/2).

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A098052 T(n,k) counts the solid partitions of n that can be extended to a solid partition of n+1 in exactly (k+3) ways. Equivalently, the number of solid partitions of n that have exactly k+3 partitions of n+1 majoring them.

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A318966 Expansion of e.g.f. Product_{i>=1, j>=1, k>=1} 1/(1 - x^(i*j*k))^(1/(i*j*k)).

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A304964 Expansion of 1/(1 - Sum_{i>=1, j>=1, k>=1, l>=1} x^(ijk*l)).

A321240 Expansion of Product_{i>=1, j>=1, k>=1, l>=1} (1 + x^(ijkl))/(1 - x^(ijkl)).

A318966 Expansion of e.g.f. Product_{i>=1, j>=1, k>=1} 1/(1 - x^(ijk))^(1/(ijk)).

A321302 Expansion of Product_{i>=1, j>=1, k>=1, l>=1} (1 - x^(ijkl))/(1 + x^(ijkl)).