A328373 Expansion of Sum_{k>=1} x^(k^2) * (1 + x^(2*k^2)) / (1 - x^(2*k^2))^2.

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, 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, 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

Offset: 1

Author

Ilya Gutkovskiy, Oct 14 2019

Keywords

• nonn
• mult
• easy

Comments

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

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000593, A010052, A035316, A036554 (positions of 0's), A056911 (fixed points), A076752, A193356, A328372.

Programs

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

Mathematica

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
Table[DivisorSum[n, # &, OddQ[#] && IntegerQ[(n/#)^(1/2)] &], {n, 1, 81}]
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 *)

PARI

a(n) = sumdiv(n, d, if ((d%2) && issquare(n/d), d)); \\ Michel Marcus, Oct 14 2019

Formula

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

G.f.: Sum_{i>=1} Sum_{j>=1} phi(i) * x^(i*j^2) / (1 + x^(i*j^2)).

Dirichlet g.f.: (1 - 2^(1 - s)) * zeta(s-1) * zeta(2*s).

a(n) = Sum_{d|n} A193356(d) * A010052(n/d).

Sum_{k=1..n} a(k) ~ Pi^4 * n^2 / 360. - Vaclav Kotesovec, Oct 14 2019

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