This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A328373 #17 Sep 08 2022 08:46:24
%S A328373 1,0,3,1,5,0,7,0,10,0,11,3,13,0,15,1,17,0,19,5,21,0,23,0,26,0,30,7,29,
%T A328373 0,31,0,33,0,35,10,37,0,39,0,41,0,43,11,50,0,47,3,50,0,51,13,53,0,55,
%U A328373 0,57,0,59,15,61,0,70,1,65,0,67,17,69,0,71,0,73,0,78,19,77,0,79,5,91
%N A328373 Expansion of Sum_{k>=1} x^(k^2) * (1 + x^(2*k^2)) / (1 - x^(2*k^2))^2.
%C A328373 Sum of odd divisors d of n such that n/d is square.
%H A328373 Amiram Eldar, <a href="/A328373/b328373.txt">Table of n, a(n) for n = 1..10000</a>
%F A328373 G.f.: Sum_{k>=1} (2*k - 1) * (theta_3(x^(2*k - 1)) - 1) / 2.
%F A328373 G.f.: Sum_{i>=1} Sum_{j>=1} phi(i) * x^(i*j^2) / (1 + x^(i*j^2)).
%F A328373 Dirichlet g.f.: (1 - 2^(1 - s)) * zeta(s-1) * zeta(2*s).
%F A328373 a(n) = Sum_{d|n} A193356(d) * A010052(n/d).
%F A328373 Sum_{k=1..n} a(k) ~ Pi^4 * n^2 / 360. - _Vaclav Kotesovec_, Oct 14 2019
%F A328373 Multiplicative with a(2^e) = 0 if e is odd, and 1 if e is even, and for p > 2, a(p^e) = (p^(e + 2) - p)/(p^2 - 1) if e is odd, and (p^(e + 2) - 1)/(p^2 - 1) if e is even. - _Amiram Eldar_, Oct 16 2020
%t A328373 nmax = 81; CoefficientList[Series[Sum[x^(k^2) (1 + x^(2 k^2))/(1 - x^(2 k^2))^2, {k, 1, Floor[Sqrt[nmax]] + 1}], {x, 0, nmax}], x] // Rest
%t A328373 Table[DivisorSum[n, # &, OddQ[#] && IntegerQ[(n/#)^(1/2)] &], {n, 1, 81}]
%t A328373 f[p_, e_] := If[p == 2, Boole @ EvenQ[e], If[EvenQ[e], (p^(e + 2) - 1)/(p^2 - 1), (p^(e + 2) - p)/(p^2 - 1)]]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* _Amiram Eldar_, Oct 16 2020 *)
%o A328373 (PARI) a(n) = sumdiv(n, d, if ((d%2) && issquare(n/d), d)); \\ _Michel Marcus_, Oct 14 2019
%o A328373 (Magma) a:=[];for n in [1..81] do v:=[d:d in Divisors(n)| IsOdd(d) and IsSquare(n div d)]; if #v ne 0 then Append(~a,&+v); else Append(~a,0); end if; end for; a; // _Marius A. Burtea_, Oct 14 2019
%Y A328373 Cf. A000593, A010052, A035316, A036554 (positions of 0's), A056911 (fixed points), A076752, A193356, A328372.
%K A328373 nonn,mult,easy
%O A328373 1,3
%A A328373 _Ilya Gutkovskiy_, Oct 14 2019