A291108 Expansion of Sum_{k>=2} k^2*x^(2*k)/(1 - x^k).

0, 0, 0, 4, 0, 13, 0, 20, 9, 29, 0, 65, 0, 53, 34, 84, 0, 130, 0, 145, 58, 125, 0, 273, 25, 173, 90, 265, 0, 399, 0, 340, 130, 293, 74, 614, 0, 365, 178, 609, 0, 735, 0, 625, 340, 533, 0, 1105, 49, 754, 298, 865, 0, 1183, 146, 1113, 370, 845, 0, 1859, 0, 965, 580, 1364, 194, 1743, 0, 1465, 538, 1599, 0, 2550, 0, 1373, 884

Offset: 1

Author

Ilya Gutkovskiy, Aug 17 2017

Keywords

• nonn

Comments

Sum of squares of divisors of n except 1 and n^2 (sum of squares of nontrivial divisors of n).

Examples

			a(6) = 13 because 6 has 4 divisors {1, 2, 3, 6} among which 2 are nontrivial {2, 3} and 2^2 + 3^2 = 13.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000
• Index entries for sequences related to sums of divisors

Crossrefs

Cf. A000290, A001157, A001248, A001358, A005063, A008578, A027751, A048050, A053807, A067558, A070824.

Programs

Mathematica

nmax = 75; Rest[CoefficientList[Series[Sum[k^2 x^(2 k)/(1 - x^k), {k, 2, nmax}], {x, 0, nmax}], x]]
Join[{0}, Table[DivisorSigma[2, n] - n^2 - 1, {n, 2, 75}]]

PARI

A291108(n) = sumdiv(n,d,if((1==d)||(n==d),0,d^2)); \\ Antti Karttunen, Jan 22 2025

Formula

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

a(n) = A001157(n) - A000290(n) - 1 for n > 1.

a(n) = A067558(n) - 1 for n > 1.

a(n) = A005063(n) if n is a semiprime (A001358).

a(n) = 0 if n is a prime or 1 (A008578).

a(n) = n if n is a square of prime (A001248).

a(p^k) = (p^(2*k) - p^2)/(p^2 - 1) for p is a prime and k > 0.