A373440 Denominator of sum of reciprocals of square divisors of n.

1, 1, 1, 4, 1, 1, 1, 4, 9, 1, 1, 4, 1, 1, 1, 16, 1, 9, 1, 4, 1, 1, 1, 4, 25, 1, 9, 4, 1, 1, 1, 16, 1, 1, 1, 18, 1, 1, 1, 4, 1, 1, 1, 4, 9, 1, 1, 16, 49, 25, 1, 4, 1, 9, 1, 4, 1, 1, 1, 4, 1, 1, 9, 64, 1, 1, 1, 4, 1, 1, 1, 18, 1, 1, 25, 4, 1, 1, 1, 16, 81, 1, 1, 4, 1

Offset

1,4

Links

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

Formula

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

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

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 // Denominator
f[p_, e_] := (p^2 - p^(-2*Floor[e/2]))/(p^2-1); a[1] = 1; a[n_] := Denominator[Times @@ f @@@ FactorInteger[n]]; Array[a, 100] (* Amiram Eldar, Jun 26 2024 *)

Prog

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

Crossrefs

Cf. A007407, A007947, A017666, A017668, A035316, A332881, A373439 (numerators).

Sequence in context: A335324 A366245 A083730 * A358272 A008833 A162400

Adjacent sequences: A373437 A373438 A373439 * A373441 A373442 A373443

Keyword

nonn,frac

Author

Ilya Gutkovskiy, Jun 05 2024

Status

approved