A373440 - OEIS

%I #15 Jun 26 2024 06:02:22

%S 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,

%T 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,

%U 1,1,4,1,1,1,18,1,1,25,4,1,1,1,16,81,1,1,4,1

%N Denominator of sum of reciprocals of square divisors of n.

%H Amiram Eldar, <a href="/A373440/b373440.txt">Table of n, a(n) for n = 1..10000</a>

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

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

%e 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, ...

%t nmax = 85; CoefficientList[Series[Sum[x^(k^2)/(k^2 (1 - x^(k^2))), {k, 1, nmax}], {x, 0, nmax}], x] // Rest // Denominator

%t 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 *)

%o (PARI) a(n) = denominator(sumdiv(n, d, if (issquare(d), 1/d))); \\ _Michel Marcus_, Jun 05 2024

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

%K nonn,frac

%O 1,4

%A _Ilya Gutkovskiy_, Jun 05 2024