A092348 -id:A092348 - cp's OEIS frontend

A363601 Number of partitions of n where there are k^2 - 1 kinds of parts k.

1, 0, 3, 8, 21, 48, 126, 288, 693, 1568, 3570, 7896, 17417, 37632, 80823, 171192, 359733, 747936, 1543192, 3155760, 6407037, 12909024, 25835649, 51359136, 101470854, 199264128, 389096028, 755591256, 1459643343, 2805471984, 5366161740, 10216161336, 19362398580

Offset: 0

Seiichi Manyama, Jun 10 2023

Cf. A005380, A023871, A052847, A092348, A120844, A217093, A253289, A255271, A255802, A255803, A255835, A255836, A363602.

with(numtheory):
series(exp(add((sigma[3](k) - sigma[1](k))*x^k/k, k = 1..50)), x, 51):
seq(coeftayl(%, x = 0, n), n = 0..50); # Peter Bala, Jan 16 2025

my(N=40, x='x+O('x^N)); Vec(1/prod(k=1, N, (1-x^k)^(k^2-1)))

G.f.: 1/Product_{k>=1} (1-x^k)^(k^2-1).
a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} A092348(k) * a(n-k).
G.f.: exp(Sum_{k >= 1} (sigma_3(k) - sigma_1(k))*x^k/k) = 1 + 3*x^2 + 8*x^3 + 21*x^4 + 48*x^5 + .... - Peter Bala, Jan 16 2025

Name suggested by Joerg Arndt, Jun 11 2023

A363604 Expansion of Sum_{k>0} x^(2*k)/(1-x^k)^4.

Original entry on oeis.org

0, 1, 4, 11, 20, 40, 56, 95, 124, 186, 220, 336, 364, 512, 584, 775, 816, 1129, 1140, 1526, 1600, 1992, 2024, 2720, 2620, 3290, 3400, 4176, 4060, 5280, 4960, 6231, 6208, 7362, 7216, 9195, 8436, 10280, 10248, 12270, 11480, 14432, 13244, 16192, 15884, 18240

Offset: 1

Seiichi Manyama, Jun 11 2023

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000292, A032741, A065608, A069153, A363605, A363606.
Cf. A059358, A363607, A363611.
Cf. A000203, A001158, A013662, A092348.

a[n_] := (DivisorSigma[3, n] - DivisorSigma[1, n])/6; Array[a, 50] (* Amiram Eldar, Jul 25 2023 *)

my(N=50, x='x+O('x^N)); concat(0, Vec(sum(k=1, N, x^(2*k)/(1-x^k)^4)))

a(n) = my(f = factor(n)); (sigma(f, 3) - sigma(f))/6; \\ Amiram Eldar, Dec 30 2024

a(n) = (sigma_3(n) - sigma(n))/6 = A092348(n)/6.
G.f.: Sum_{k>0} binomial(k+1,3) * x^k/(1 - x^k).
From Amiram Eldar, Dec 30 2024: (Start)
Dirichlet g.f.: zeta(s) * (zeta(s-3) - zeta(s-1)) / 6.
Sum_{k=1..n} a(k) ~ (zeta(4)/24) * n^4. (End)

A086666 a(n) = sigma_2(n) - sigma_1(n).

Original entry on oeis.org

0, 2, 6, 14, 20, 38, 42, 70, 78, 112, 110, 182, 156, 226, 236, 310, 272, 416, 342, 504, 468, 574, 506, 790, 620, 808, 780, 994, 812, 1228, 930, 1302, 1172, 1396, 1252, 1820, 1332, 1750, 1644, 2120, 1640, 2404, 1806, 2478, 2288, 2578, 2162, 3286, 2394, 3162

Offset: 1

Jon Perry, Jul 27 2003

Total area of all distinct L X W rectangles such that s + t = n, 1 <= s <= t, s | n, L = n/s and W = t/s. - Wesley Ivan Hurt, Aug 01 2025

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Joerg Arndt, On computing the generalized Lambert series, arXiv:1202.6525v3 [math.CA], (2012).

Cf. A000203, A001157, A002117, A052847, A069153, A092348.

Table[DivisorSigma[2,n]-DivisorSigma[1,n],{n,50}] (* Harvey P. Dale, Aug 01 2020 *)

PARI
```
a(n) = sigma(n,2)-sigma(n,1);
```

a(n) = my(f = factor(n)); sigma(f, 2) - sigma(f); \\ Amiram Eldar, Jan 01 2025

G.f.: Sum_{n>=1} n*(n-1) * x^n/(1-x^n). - Joerg Arndt, Jan 30 2011
L.g.f.: -log(Product_{k>=1} (1 - x^k)^(k-1)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, May 21 2018
From Peter Bala, Jan 21 2021: (Start)
a(n) = 2*A069153(n).
G.f.: A(x) = Sum_{n >= 1} 2*x^(2*n)/(1 - x^n)^3.
A faster converging series: A(x) = Sum_{n >= 1} x^(n^2)*( n*(n-1)*x^(3*n) - (n^2 + n - 2)*x^(2*n) + n*(3 - n)*x^n + n*(n - 1) )/(1 - x^n)^3 - differentiate equation 5 in Arndt twice w.r.t x and set x = 1. (End)
From Amiram Eldar, Jan 01 2025: (Start)
Dirichlet g.f.: zeta(s) * (zeta(s-2) - zeta(s-1)).
Sum_{k=1..n} a(k) ~ (zeta(3)/3) * n^3. (End)
a(n) = Sum_{d|n} d*(d-1). - Wesley Ivan Hurt, Aug 01 2025

A379813 a(n) = sigma_1(n) * sigma_3(n).

Original entry on oeis.org

1, 27, 112, 511, 756, 3024, 2752, 8775, 9841, 20412, 15984, 57232, 30772, 74304, 84672, 145111, 88452, 265707, 137200, 386316, 308224, 431568, 292032, 982800, 488281, 830844, 817600, 1406272, 731700, 2286144, 953344, 2359287, 1790208, 2388204, 2080512, 5028751

Offset: 1

Amiram Eldar, Jan 03 2025

See A379812 for more links and Ramanujan's general formula.

Srinivasa Ramanujan, Collected papers, edited by G. H. Hardy et al., Chelsea, 1962, chapter 17, pp. 133-135.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Srinivasa Ramanujan, Some formulae in the analytic theory of numbers, Messenger of Mathematics, Vol. 45 (1916), pp. 81-84.

Cf. A000203, A001158, A013663, A072861, A091258, A091259, A092348, A092345, A356533, A356534, A379812, A379814.

a[n_] := Times @@ DivisorSigma[{1, 3}, n]; Array[a, 50]

a(n) = {my(f = factor(n)); sigma(f) * sigma(f, 3);}

a(n) = A000203(n) * A001158(n).
Multiplicative with a(p^e) = (p^(e+1)-1) * (p^(3*e+3)-1) / ((p-1) * (p^3-1)).
Dirichlet g.f.: zeta(s) * zeta(s-1) * zeta(s-3) * zeta(s-4) / zeta(2*s-4).
Sum_{k=1..n}  a(k) ~ c * n^5 / 5, where c = 7 * zeta(5) / 4 = 1.81462357150089737107... .

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A363601 Number of partitions of n where there are k^2 - 1 kinds of parts k.

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A363604 Expansion of Sum_{k>0} x^(2*k)/(1-x^k)^4.

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A086666 a(n) = sigma_2(n) - sigma_1(n).

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A379813 a(n) = sigma_1(n) * sigma_3(n).

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