A373439 Numerator of sum of reciprocals of square divisors of n.

1, 1, 1, 5, 1, 1, 1, 5, 10, 1, 1, 5, 1, 1, 1, 21, 1, 10, 1, 5, 1, 1, 1, 5, 26, 1, 10, 5, 1, 1, 1, 21, 1, 1, 1, 25, 1, 1, 1, 5, 1, 1, 1, 5, 10, 1, 1, 21, 50, 26, 1, 5, 1, 10, 1, 5, 1, 1, 1, 5, 1, 1, 10, 85, 1, 1, 1, 5, 1, 1, 1, 25, 1, 1, 26, 5, 1, 1, 1, 21, 91, 1, 1, 5, 1

Offset: 1

Author

Ilya Gutkovskiy, Jun 05 2024

Keywords

• nonn
• frac

Examples

			1, 1, 1, 5/4, 1, 1, 1, 5/4, 10/9, 1, 1, 5/4, 1, 1, 1, 21/16, 1, 10/9, 1, 5/4, 1, 1, 1, 5/4, 26/25, ...
		

Links

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

Crossrefs

Cf. A007406, A017665, A017667, A028235, A035316, A332880, A373440 (denominators).

Programs

Mathematica

nmax = 85; CoefficientList[Series[Sum[x^(k^2)/(k^2 (1 - x^(k^2))), {k, 1, nmax}], {x, 0, nmax}], x] // Rest // Numerator
f[p_, e_] := (p^2 - p^(-2*Floor[e/2]))/(p^2-1); a[1] = 1; a[n_] := Numerator[Times @@ f @@@ FactorInteger[n]]; Array[a, 100] (* Amiram Eldar, Jun 26 2024 *)

PARI

a(n) = numerator(sumdiv(n, d, if (issquare(d), 1/d))); \\ Michel Marcus, Jun 05 2024

Formula

Numerators of coefficients in expansion of Sum_{k>=1} x^(k^2)/(k^2*(1 - x^(k^2))).

a(n) is the numerator of Sum_{d^2|n} 1/d^2.

From Amiram Eldar, Jun 26 2024: (Start)

Let f(n) = a(n)/A373440(n). Then:

f(n) is multiplicative with f(p^e) = (p^2 - p^(-2*floor(e/2)))/(p^2-1).

Dirichlet g.f. of f(n): zeta(s) * zeta(2*s+2).

Sum_{k=1..n} f(k) ~ zeta(4) * n. (End)