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

%I A291108 #12 Jan 22 2025 11:40:14
%S A291108 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,
%T A291108 173,90,265,0,399,0,340,130,293,74,614,0,365,178,609,0,735,0,625,340,
%U A291108 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
%N A291108 Expansion of Sum_{k>=2} k^2*x^(2*k)/(1 - x^k).
%C A291108 Sum of squares of divisors of n except 1 and n^2 (sum of squares of nontrivial divisors of n).
%H A291108 Antti Karttunen, <a href="/A291108/b291108.txt">Table of n, a(n) for n = 1..20000</a>
%H A291108 <a href="/index/Su#sums_of_divisors">Index entries for sequences related to sums of divisors</a>
%F A291108 G.f.: Sum_{k>=2} k^2*x^(2*k)/(1 - x^k).
%F A291108 a(n) = A001157(n) -  A000290(n) - 1 for n > 1.
%F A291108 a(n) = A067558(n) - 1 for n > 1.
%F A291108 a(n) = A005063(n) if n is a semiprime (A001358).
%F A291108 a(n) = 0 if n is a prime or 1 (A008578).
%F A291108 a(n) = n if n is a square of prime (A001248).
%F A291108 a(p^k) = (p^(2*k) - p^2)/(p^2 - 1) for p is a prime and k > 0.
%e A291108 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.
%t A291108 nmax = 75; Rest[CoefficientList[Series[Sum[k^2 x^(2 k)/(1 - x^k), {k, 2, nmax}], {x, 0, nmax}], x]]
%t A291108 Join[{0}, Table[DivisorSigma[2, n] - n^2 - 1, {n, 2, 75}]]
%o A291108 (PARI) A291108(n) = sumdiv(n,d,if((1==d)||(n==d),0,d^2)); \\ _Antti Karttunen_, Jan 22 2025
%Y A291108 Cf. A000290, A001157, A001248, A001358, A005063, A008578, A027751, A048050, A053807, A067558, A070824.
%K A291108 nonn
%O A291108 1,4
%A A291108 _Ilya Gutkovskiy_, Aug 17 2017