A087115 Convolution of sum of cubes of divisors with itself.

0, 1, 18, 137, 650, 2350, 6860, 17609, 39870, 83976, 162382, 301070, 522886, 885284, 1424468, 2254537, 3419448, 5143987, 7448874, 10750712, 15015872, 20948610, 28373444, 38539022, 50863150, 67454492, 87209316, 113326308, 143748766, 183759900, 229271536

Offset: 1

Author

Michael Somos, Aug 13 2003

Keywords

• nonn
• changed

Comments

Ramanujan's Eisenstein series: P(q) (see A006352), Q(q) (A004009), R(q) (A013973).

Examples

			G.f. = x^2 + 18*x^3 + 137*x^4 + 650*x^5 + 2350*x^6 + 6860*x^7 + 17609*x^8 + ...
		

References

• Jean-Pierre Serre, A Course in Arithmetic, Springer-Verlag, 1973, Chapter VII, Section 4., p. 93.

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A004009.

Cf. A001158 (sigma_3), A013955 (sigma_7). [Ridouane Oudra, Apr 22 2020]

Programs

Maple

with(numtheory); f:=n->add( sigma[3](k)*sigma[3](n-k),k=1..n-1);

Mathematica

a[ n_] := If[ n < 1, 0, (DivisorSigma[ 7, n] - DivisorSigma[ 3, n]) / 120]; (* Michael Somos, Oct 08 2017 *)

PARI

{a(n) = if( n<1, 0, (sigma(n, 7) - sigma(n, 3)) / 120)};

PARI

{a(n) = if( n<1, 0, sum(m=1, n-1, sigma(m, 3) * sigma(n-m, 3)))};

Formula

G.f.: (Sum_{k>0} k^3 * x^k / (1 - x^k))^2.

a(n) = (sigma_7(n) - sigma_3(n)) / 120.

G.f.: ((Q(x) - 1) / 240)^2 where Q() is a Ramanujan Eisenstein series.

Dirichlet g.f.: zeta(s) * (zeta(s-7) - zeta(s-3)) / 120. - Amiram Eldar, Jan 11 2025

Sum_{k=1..n} a(k) ~ Pi^8 * n^8 / 9072000. - Vaclav Kotesovec, Aug 20 2025