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Numerator of sum of reciprocals of square divisors of n.
2

%I #17 Jun 26 2024 06:01:58

%S 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,

%T 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,

%U 10,85,1,1,1,5,1,1,1,25,1,1,26,5,1,1,1,21,91,1,1,5,1

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

%H Amiram Eldar, <a href="/A373439/b373439.txt">Table of n, a(n) for n = 1..10000</a>

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

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

%F From _Amiram Eldar_, Jun 26 2024: (Start)

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

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

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

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

%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 // Numerator

%t 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 *)

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

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

%K nonn,frac

%O 1,4

%A _Ilya Gutkovskiy_, Jun 05 2024