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

1, 4, 9, 17, 25, 36, 49, 68, 82, 100, 121, 153, 169, 196, 225, 273, 289, 328, 361, 425, 441, 484, 529, 612, 626, 676, 738, 833, 841, 900, 961, 1092, 1089, 1156, 1225, 1394, 1369, 1444, 1521, 1700, 1681, 1764, 1849, 2057, 2050, 2116, 2209, 2457, 2402, 2504, 2601, 2873, 2809, 2952, 3025

Offset: 1

Author

Ilya Gutkovskiy, Oct 10 2019

Keywords

• nonn
• mult

Comments

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

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000

Crossrefs

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

Programs

Magma

[&+[d^2:d in Divisors(n)| IsSquare(n div d)]:n in [1..55]]; // Marius A. Burtea, Oct 10 2019

Maple

a:= n-> add((n/d)^2, d=select(issqr, numtheory[divisors](n))):
seq(a(n), n=1..60);  # Alois P. Heinz, Oct 11 2019

Mathematica

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

PARI

a(n) = sumdiv(n, d, if (issquare(n/d), d^2)); \\ Michel Marcus, Oct 12 2019

Formula

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

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

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

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

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

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

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