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A328271 Expansion of Sum_{k>=1} x^(k^2) * (1 + x^(k^2)) / (1 - x^(k^2))^3.

%I A328271 #24 Sep 08 2022 08:46:24
%S A328271 1,4,9,17,25,36,49,68,82,100,121,153,169,196,225,273,289,328,361,425,
%T A328271 441,484,529,612,626,676,738,833,841,900,961,1092,1089,1156,1225,1394,
%U A328271 1369,1444,1521,1700,1681,1764,1849,2057,2050,2116,2209,2457,2402,2504,2601,2873,2809,2952,3025
%N A328271 Expansion of Sum_{k>=1} x^(k^2) * (1 + x^(k^2)) / (1 - x^(k^2))^3.
%C A328271 Sum of squares of divisors d of n such that n/d is square.
%H A328271 Alois P. Heinz, <a href="/A328271/b328271.txt">Table of n, a(n) for n = 1..10000</a>
%F A328271 G.f.: Sum_{k>=1} k^2 * (theta_3(x^k) - 1)/2.
%F A328271 Dirichlet g.f.: zeta(2*s) * zeta(s-2).
%F A328271 a(n) = Sum_{d|n} A010052(n/d) * d^2.
%F A328271 a(n) = Sum_{d|n} |A076792(d)|.
%F A328271 a(p) = p^2, where p is prime.
%F A328271 Sum_{k=1..n} a(k) ~ Pi^6 * n^3 / 2835. - _Vaclav Kotesovec_, Oct 11 2019
%F A328271 Multiplicative with a(p^e) = Sum_{i=0..floor(e/2)} p^(2*e-4*i) for prime p, i.e., a(p^(2*e)) = (p^(4*e+4)-1)/(p^4-1) and a(p^(2*e+1)) = p^2 * (p^(4*e+4)-1)/(p^4-1) for prime p. - _Werner Schulte_, Jul 24 2021
%p A328271 a:= n-> add((n/d)^2, d=select(issqr, numtheory[divisors](n))):
%p A328271 seq(a(n), n=1..60);  # _Alois P. Heinz_, Oct 11 2019
%t A328271 nmax = 55; CoefficientList[Series[Sum[x^(k^2) (1 + x^(k^2))/(1 - x^(k^2))^3, {k, 1, Floor[Sqrt[nmax]] + 1}], {x, 0, nmax}], x] // Rest
%t A328271 Table[DivisorSum[n, #^2 &, IntegerQ[Sqrt[n/#]] &], {n, 1, 55}]
%o A328271 (Magma) [&+[d^2:d in Divisors(n)| IsSquare(n div d)]:n in [1..55]]; // _Marius A. Burtea_, Oct 10 2019
%o A328271 (PARI) a(n) = sumdiv(n, d, if (issquare(n/d), d^2)); \\ _Michel Marcus_, Oct 12 2019
%Y A328271 Cf. A000290, A001157, A010052, A076752, A076792.
%K A328271 nonn,mult
%O A328271 1,2
%A A328271 _Ilya Gutkovskiy_, Oct 10 2019