A328372 Expansion of Sum_{k>=1} x^(k^2) / (1 - x^(2*k^2)).

1, 0, 1, 1, 1, 0, 1, 0, 2, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 2, 0, 2, 1, 1, 0, 1, 0, 1, 0, 1, 2, 1, 0, 1, 0, 1, 0, 1, 1, 2, 0, 1, 1, 2, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 2, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 2, 1, 1, 0, 1, 1, 3, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1

Offset: 1

Author

Ilya Gutkovskiy, Oct 14 2019

Keywords

• nonn
• mult
• easy

Comments

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

Inverse Moebius transform of A143259.

Links

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

Crossrefs

Cf. A000035, A001227, A010052, A036554 (positions of 0's), A046951, A143259, A328373.

Programs

Magma

[#[d:d in Divisors(n)| IsOdd(d) and IsSquare(n div d)]:n in [1..90]]; // Marius A. Burtea, Oct 14 2019

Mathematica

nmax = 91; CoefficientList[Series[Sum[x^(k^2)/(1 -  x^(2 k^2)), {k, 1, Floor[Sqrt[nmax]] + 1}], {x, 0, nmax}], x] // Rest
Table[DivisorSum[n, 1 &, OddQ[#] && IntegerQ[(n/#)^(1/2)] &], {n, 1, 91}]
Table[DivisorSum[n, (-1)^PrimeOmega[#] Mod[DivisorSigma[1, #], 2] &], {n, 1, 91}]
f[p_, e_] := If[p == 2, Boole @ EvenQ[e], Ceiling[(e + 1)/2]]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Oct 16 2020 *)

PARI

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

Formula

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

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

a(n) = Sum_{d|n} A000035(d) * A010052(n/d).

Sum_{k=1..n} a(k) ~ Pi^2 * n / 12 + (sqrt(2)-1) * zeta(1/2) * sqrt(n/2). - 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) = ceiling((e+1)/2). - Amiram Eldar, Oct 16 2020