A280451 -id:A280451 - cp's OEIS frontend

A004101 Number of partitions of n of the form a_1b_1^2 + a_2b_2^2 + ...; number of semisimple rings with p^n elements for any prime p.

1, 1, 2, 3, 6, 8, 13, 18, 29, 40, 58, 79, 115, 154, 213, 284, 391, 514, 690, 900, 1197, 1549, 2025, 2600, 3377, 4306, 5523, 7000, 8922, 11235, 14196, 17777, 22336, 27825, 34720, 43037, 53446, 65942, 81423, 100033, 122991, 150481, 184149, 224449, 273614, 332291

Offset: 0

N. J. A. Sloane

The number of semisimple rings with p^n elements does not depend on the prime number p. - Paul Laubie, Mar 05 2024

			4 = 4*1^2 = 1*2^2 = 3*1^2 + 1*1^2 = 2*1^2 + 2*1^2 = 2*1^2 + 1*1^2 + 1*1^2 = 1*1^2 + 1*1^2 + 1*1^2 + 1*1^2.

J. Knopfmacher, Abstract Analytic Number Theory. North-Holland, Amsterdam, 1975, p. 293.
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..10000 (terms 0..1000 from Alois P. Heinz)
Gert Almkvist, Asymptotics of various partitions, arXiv:math/0612446 [math.NT], 2006 (section 6).
I. G. Connell, A number theory problem concerning finite groups and rings, Canad. Math. Bull, 7 (1964), 23-34.
I. G. Connell, Letter to N. J. A. Sloane, no date
N. J. A. Sloane, Transforms

Cf. A006171, A038538, A280451, A280661, A280662.

with(numtheory):
a:= proc(n) option remember;
      `if`(n=0, 1, add(add(d* mul(1+iquo(i[2], 2),
      i=ifactors(d)[2]), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Nov 26 2013
sqd:=proc(n) local t1,d; t1:=0; for d in divisors(n) do if (n mod d^2) = 0 then t1:=t1+1; fi; od; t1; end; # A046951
t2:=mul( 1/(1-x^n)^sqd(n),n=1..65); series(t2,x,60); seriestolist(%); # N. J. A. Sloane, Jun 24 2015

max = 45; A046951 = Table[Sum[Floor[n/k^2], {k, n}], {n, 0, max}] // Differences; f = Product[1/(1-x^n)^A046951[[n]], {n, 1, max}]; CoefficientList[Series[f, {x, 0, max}], x] (* Jean-François Alcover, Feb 11 2014 *)
nmax = 50; CoefficientList[Series[Product[1/(1 - x^(j*k^2)), {k, 1, Floor[Sqrt[nmax]] + 1}, {j, 1, Floor[nmax/k^2] + 1}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 03 2017 *)

N=66; x='x+O('x^N); gf=1/prod(j=1,N, eta(x^(j^2))); Vec(gf) /* Joerg Arndt, May 03 2008 */

EULER transform of A046951.

a(n) ~ exp(Pi^2 * sqrt(n) / 3 + sqrt(3/(2*Pi)) * Zeta(1/2) * Zeta(3/2) * n^(1/4) - 9 * Zeta(1/2)^2 * Zeta(3/2)^2 / (16*Pi^3)) * Pi^(3/4) / (sqrt(2) * 3^(1/4) * n^(5/8)) [Almkvist, 2006]. - Vaclav Kotesovec, Jan 03 2017

More terms, formula and better description from Christian G. Bower, Nov 15 1999

Name clarified by Paul Laubie, Mar 05 2024

A280664 G.f.: Product_{k>=1, j>=1} (1 + x^(j*k^4)).

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 18, 22, 27, 33, 39, 47, 56, 66, 79, 93, 109, 128, 150, 175, 204, 237, 274, 318, 367, 423, 487, 559, 641, 734, 839, 957, 1091, 1241, 1410, 1601, 1814, 2053, 2322, 2622, 2957, 3334, 3752, 4218, 4740, 5318, 5962, 6679

Offset: 0

Vaclav Kotesovec, Jan 07 2017

nonn

In general, if m>=3 and g.f. = Product_{k>=1, j>=1} (1+x^(j*k^m)), then a(n, m) ~ exp(Pi*sqrt(Zeta(m)*n/3) + (2^(1/m)-1) * Pi^(-1/m) * Gamma(1+1/m) * Zeta(1+1/m) * Zeta(1/m) * (3*n/Zeta(m))^(1/(2*m))) * Zeta(m)^(1/4) / (2^(5/4) * 3^(1/4) * n^(3/4)).

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A107742 (m=1), A280451 (m=2), A280663 (m=3).

Cf. A280662.

nmax = 100; CoefficientList[Series[Product[1+x^(j*k^4), {k, 1, Floor[nmax^(1/4)]+1}, {j, 1, Floor[nmax/k^4]+1}], {x, 0, nmax}], x]

a(n) ~ exp(Pi^3 * sqrt(n/30)/3 + 2^(-15/8) * 3^(3/8) * 5^(1/8) * (2^(1/4)-1) * Pi^(-3/4) * Gamma(1/4) * Zeta(5/4) * Zeta(1/4) * n^(1/8)) * Pi / (2^(3/2) * 3^(3/4) * 5^(1/4) * n^(3/4)).

A320235 G.f.: Product_{k>=1, j>=1} (1 + x^(k*j))^2.

Original entry on oeis.org

1, 2, 5, 12, 24, 48, 94, 172, 310, 550, 946, 1602, 2679, 4394, 7123, 11424, 18082, 28344, 44039, 67754, 103412, 156660, 235489, 351602, 521650, 768998, 1127100, 1642946, 2381929, 3436028, 4932998, 7049004, 10028422, 14207122, 20044327, 28169528, 39439899

Offset: 0

Vaclav Kotesovec, Oct 08 2018

nonn

Self-convolution of A107742.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A107742, A280451, A320236.

nmax = 50; CoefficientList[Series[Product[(1+x^(k*j))^2, {k, 1, nmax}, {j, 1, Floor[nmax/k] + 1}], {x, 0, nmax}], x]

Conjecture: log(a(n)) ~ Pi*sqrt(n*log(n)/3).

A280663 G.f.: Product_{k>=1, j>=1} (1 + x^(j*k^3)).

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 11, 14, 17, 21, 26, 32, 39, 47, 57, 68, 81, 97, 115, 136, 162, 190, 223, 263, 306, 357, 417, 483, 561, 650, 750, 866, 997, 1145, 1315, 1507, 1725, 1971, 2250, 2564, 2917, 3318, 3766, 4270, 4840, 5475, 6188, 6990, 7881, 8881

Offset: 0

Vaclav Kotesovec, Jan 07 2017

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A107742, A280451, A280664.

nmax = 100; CoefficientList[Series[Product[1+x^(j*k^3), {k, 1, Floor[nmax^(1/3)]+1}, {j, 1, Floor[nmax/k^3]+1}], {x, 0, nmax}], x]

a(n) ~ exp(Pi*sqrt(Zeta(3)*n/3) + (2^(1/3)-1) * Pi^(-1/3) * Gamma(4/3) * Zeta(4/3) * Zeta(1/3) * (3*n/Zeta(3))^(1/6)) * Zeta(3)^(1/4) / (2^(5/4) * 3^(1/4) * n^(3/4)).

A327745 Expansion of Product_{i>=1, j>=1} (1 + x^(ij(j + 1)/2)).

Original entry on oeis.org

1, 1, 1, 3, 3, 4, 8, 9, 11, 19, 23, 28, 42, 51, 62, 89, 108, 130, 178, 215, 260, 344, 413, 496, 639, 766, 916, 1155, 1380, 1641, 2040, 2426, 2870, 3520, 4166, 4912, 5960, 7023, 8246, 9911, 11634, 13610, 16224, 18972, 22111, 26183, 30507, 35430, 41698

Offset: 0

Ilya Gutkovskiy, Sep 23 2019

nonn

Weigh transform of A007862.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A007862, A280451, A327744.

nmax = 48; CoefficientList[Series[Product[(1 + x^k)^Length[Select[Divisors[k], IntegerQ[Sqrt[8 # + 1]] &]], {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d Length[Select[Divisors[d], IntegerQ[Sqrt[8 # + 1]] &]], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 48}]
nmax = 50; CoefficientList[Series[Product[QPochhammer[-1, x^(k*(k + 1)/2)]/2, {k, 1, Sqrt[2*nmax]}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 24 2019 *)

G.f.: Product_{k>=1} (1 + x^k)^A007862(k).

A280452 G.f.: Product_{k>=1, j>=1} (1 + x^(j^2*k^2)).

Original entry on oeis.org

1, 1, 0, 0, 2, 2, 0, 0, 1, 3, 2, 0, 0, 4, 4, 0, 3, 5, 3, 1, 6, 6, 2, 2, 3, 11, 9, 1, 0, 16, 16, 0, 3, 11, 15, 7, 10, 10, 14, 14, 11, 23, 19, 9, 6, 36, 32, 4, 5, 31, 46, 18, 16, 26, 48, 36, 25, 35, 38, 36, 20, 60, 60, 28, 20, 82, 98, 30, 31, 65, 104, 64, 40

Offset: 0

Vaclav Kotesovec, Jan 03 2017

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A107742, A280451, A280453.

nmax = 100; CoefficientList[Series[Product[(1 + x^(j^2*k^2)), {k, 1, Floor[Sqrt[nmax]]+1}, {j, 1, Floor[Sqrt[nmax/k^2]] + 1}], {x, 0, nmax}], x]

A327738 Expansion of 1 / (1 - Sum_{i>=1, j>=1} x^(i*j^2)).

Original entry on oeis.org

1, 1, 2, 4, 9, 18, 37, 76, 158, 326, 672, 1386, 2862, 5906, 12187, 25148, 51900, 107103, 221023, 456110, 941256, 1942423, 4008481, 8272094, 17070712, 35227975, 72698206, 150023632, 309596255, 638898274, 1318462339, 2720844607, 5614870612, 11587126980

Offset: 0

Ilya Gutkovskiy, Sep 23 2019

nonn

Invert transform of A046951.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A004101, A046951, A129921, A280451.

a:= proc(n) option remember; `if`(n<1, 1, add(a(n-i)*
      nops(select(issqr, numtheory[divisors](i))), i=1..n))
    end:
seq(a(n), n=0..35);  # Alois P. Heinz, Sep 23 2019

nmax = 33; CoefficientList[Series[1/(1 - Sum[x^(k^2)/(1 - x^(k^2)), {k, 1, Floor[Sqrt[nmax]] + 1}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[Length[Select[Divisors[k], IntegerQ[Sqrt[#]] &]] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 33}]

G.f.: 1 / (1 - Sum_{k>=1} x^(k^2) / (1 - x^(k^2))).
G.f.: 1 / (1 - Sum_{k>=1} (theta_3(x^k) - 1) / 2), where theta_() is the Jacobi theta function.
a(0) = 1; a(n) = Sum_{k=1..n} A046951(k) * a(n-k).

A343776 G.f.: Product_{k>=1} eta(x^(k^2)).

Original entry on oeis.org

1, -1, -1, 0, -1, 2, 1, 1, -1, -1, 2, 0, -1, 0, -2, -3, -1, 2, 0, 1, 5, -2, 1, -2, 2, -1, 0, 2, 0, -1, -1, 3, -4, 4, 0, -4, -5, -2, -3, 7, 2, 1, -6, -2, 4, -2, 2, 1, 7, -5, 11, 6, 0, -1, 1, -12, -11, 5, -3, -2, -8, 9, 8, 3, 1, 2, -5, -4, 5, -11, -6, 0, 7, 7, 4, -17, 3, -5, 8, 9, -4, -1, -10, 5, -6, 24, -5, 10, 0

Offset: 0

Seiichi Manyama, Apr 29 2021

sign

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

Convolution inverse of A004101.

Cf. A010815, A280451, A288098.

my(N=99, x='x+O('x^N)); Vec(prod(k=1, sqrt(N), eta(x^(k^2))))

A327747 Expansion of Product_{i>=1, j>=1} 1 / (1 + (-x)^(i*j^2)).

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 1, 0, 1, 2, 2, 1, 1, 0, 1, 2, 1, 2, 2, 2, 1, 1, 1, 2, 3, 4, 3, 4, 4, 1, 4, 3, 4, 7, 6, 7, 6, 4, 5, 5, 7, 9, 9, 9, 8, 7, 7, 7, 10, 14, 13, 12, 14, 10, 12, 16, 13, 20, 19, 20, 20, 16, 18, 20, 22, 26, 27, 27, 28, 23, 26, 25, 31, 38, 36, 40

Offset: 0

Ilya Gutkovskiy, Sep 23 2019

nonn

Cf. A004101, A046951, A280451.

nmax = 75; CoefficientList[Series[Product[1/(1 + (-x)^k)^Length[Select[Divisors[k], IntegerQ[Sqrt[#]] &]], {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[(-1)^k Sum[(-1)^(k/d) d Length[Select[Divisors[d], IntegerQ[Sqrt[#]] &]], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 75}]

G.f.: Product_{k>=1} 1 / (1 + (-x)^k)^A046951(k).

cp's OEIS Frontend

A004101 Number of partitions of n of the form a_1*b_1^2 + a_2*b_2^2 + ...; number of semisimple rings with p^n elements for any prime p.

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A280664 G.f.: Product_{k>=1, j>=1} (1 + x^(j*k^4)).

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A320235 G.f.: Product_{k>=1, j>=1} (1 + x^(k*j))^2.

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A280663 G.f.: Product_{k>=1, j>=1} (1 + x^(j*k^3)).

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A327745 Expansion of Product_{i>=1, j>=1} (1 + x^(i*j*(j + 1)/2)).

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A280452 G.f.: Product_{k>=1, j>=1} (1 + x^(j^2*k^2)).

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A327738 Expansion of 1 / (1 - Sum_{i>=1, j>=1} x^(i*j^2)).

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A343776 G.f.: Product_{k>=1} eta(x^(k^2)).

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A004101 Number of partitions of n of the form a_1b_1^2 + a_2b_2^2 + ...; number of semisimple rings with p^n elements for any prime p.

A327745 Expansion of Product_{i>=1, j>=1} (1 + x^(ij(j + 1)/2)).