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

1,4

Links

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

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)

Example

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, ...

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 *)

Prog

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

Crossrefs

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

Sequence in context: A102280 A370239 A365403 * A035316 A293718 A068316

Adjacent sequences: A373436 A373437 A373438 * A373440 A373441 A373442

Keyword

nonn,frac

Author

Ilya Gutkovskiy, Jun 05 2024

Status

approved