A175705 Convolution square of A001157 (the sum of squared divisors).

1, 10, 45, 142, 362, 780, 1561, 2762, 4808, 7570, 12034, 17482, 26072, 35884, 50909, 67012, 92111, 116950, 155720, 193564, 250914, 304244, 389286, 461654, 578952, 680944, 839304, 970094, 1188924, 1354164, 1637145, 1858344, 2215866, 2485068

Offset: 1

Author

Michel Lagneau, Aug 10 2010

Keywords

• nonn

Links

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

Crossrefs

Cf. A001157, A328490.

Programs

Maple

with(numtheory): T:=array(1..200):for p from 1 to 200 do: liste:=divisors(p) :s2:=sum(liste[i]^2,i=1..nops(liste)):T[p] :=s2 :od : for n from 1 to 100 do: printf(`%d, `, sum (T[k]*T[n+1-k],k=1..n)):od:

Mathematica

a[n_] := Sum[DivisorSigma[2, k] * DivisorSigma[2, n + 1 - k], {k, 1, n}]; Array[a, 34] (* Amiram Eldar, Jul 31 2019 *)

Formula

a(n) = Sum_{k=1..n} A001157(k)* A001157(n+1-k).

G.f.: (1/x)*(Sum_{k>=1} k^2*x^k/(1 - x^k))^2. - Ilya Gutkovskiy, Jan 01 2017

Conjecture: Sum_{k=1..n} a(k) ~ zeta(3)^2 * n^6 / 180. - Vaclav Kotesovec, Aug 20 2025

Extensions

Definition slightly rephrased by R. J. Mathar, Aug 19 2010