A328271 - OEIS

%I #24 Sep 08 2022 08:46:24

%S 1,4,9,17,25,36,49,68,82,100,121,153,169,196,225,273,289,328,361,425,

%T 441,484,529,612,626,676,738,833,841,900,961,1092,1089,1156,1225,1394,

%U 1369,1444,1521,1700,1681,1764,1849,2057,2050,2116,2209,2457,2402,2504,2601,2873,2809,2952,3025

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

%C Sum of squares of divisors d of n such that n/d is square.

%H Alois P. Heinz, <a href="/A328271/b328271.txt">Table of n, a(n) for n = 1..10000</a>

%F G.f.: Sum_{k>=1} k^2 * (theta_3(x^k) - 1)/2.

%F Dirichlet g.f.: zeta(2*s) * zeta(s-2).

%F a(n) = Sum_{d|n} A010052(n/d) * d^2.

%F a(n) = Sum_{d|n} |A076792(d)|.

%F a(p) = p^2, where p is prime.

%F Sum_{k=1..n} a(k) ~ Pi^6 * n^3 / 2835. - _Vaclav Kotesovec_, Oct 11 2019

%F 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 a:= n-> add((n/d)^2, d=select(issqr, numtheory[divisors](n))):

%p seq(a(n), n=1..60);  # _Alois P. Heinz_, Oct 11 2019

%t 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 Table[DivisorSum[n, #^2 &, IntegerQ[Sqrt[n/#]] &], {n, 1, 55}]

%o (Magma) [&+[d^2:d in Divisors(n)| IsSquare(n div d)]:n in [1..55]]; // _Marius A. Burtea_, Oct 10 2019

%o (PARI) a(n) = sumdiv(n, d, if (issquare(n/d), d^2)); \\ _Michel Marcus_, Oct 12 2019

%Y Cf. A000290, A001157, A010052, A076752, A076792.

%K nonn,mult

%O 1,2

%A _Ilya Gutkovskiy_, Oct 10 2019