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A333694 Expansion of Sum_{k>=1} k * x^k / (1 - x^(k^2)).

%I A333694 #25 Jun 18 2022 19:18:36
%S A333694 1,3,4,5,6,9,8,9,10,13,12,16,14,17,16,17,18,21,20,25,25,25,24,25,26,
%T A333694 29,28,29,30,41,32,33,34,37,36,41,38,41,43,41,42,51,44,45,46,49,48,52,
%U A333694 50,53,52,57,54,57,61,64,61,61,60,61,62,65,64,65,66,72,68,73,70,73
%N A333694 Expansion of Sum_{k>=1} k * x^k / (1 - x^(k^2)).
%C A333694 Sum of divisors d of n such that n/d == 1 (mod d).
%F A333694 G.f.: Sum_{k>=1} k * x^k / (1 - x^(k^2)).
%p A333694 a:= n-> add(`if`(irem(n/d-1, d)=0, d, 0), d=numtheory[divisors](n)):
%p A333694 seq(a(n), n=1..80);  # _Alois P. Heinz_, Apr 04 2020
%t A333694 nmax = 70; CoefficientList[Series[Sum[k x^k/(1 - x^(k^2)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
%t A333694 Table[DivisorSum[n, # &, Mod[n/# - 1, #] == 0 &], {n, 1, 70}]
%o A333694 (PARI) A333694(n) = sumdiv(n,d,d*(0==(((n/d)-1)%d))); \\ _Antti Karttunen_, Apr 04 2020, after the second Mathematica program.
%Y A333694 Cf. A000203, A069290, A163671.
%K A333694 nonn
%O A333694 1,2
%A A333694 _Ilya Gutkovskiy_, Apr 04 2020