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A370783 a(n) is the numerator of the sum of the reciprocals of the squarefree divisors of the powerful part of n.

%I A370783 #9 Mar 02 2024 03:51:09
%S A370783 1,1,1,3,1,1,1,3,4,1,1,3,1,1,1,3,1,4,1,3,1,1,1,3,6,1,4,3,1,1,1,3,1,1,
%T A370783 1,2,1,1,1,3,1,1,1,3,4,1,1,3,8,6,1,3,1,4,1,3,1,1,1,3,1,1,4,3,1,1,1,3,
%U A370783 1,1,1,2,1,1,6,3,1,1,1,3,4,1,1,3,1,1,1
%N A370783 a(n) is the numerator of the sum of the reciprocals of the squarefree divisors of the powerful part of n.
%H A370783 Amiram Eldar, <a href="/A370783/b370783.txt">Table of n, a(n) for n = 1..10000</a>
%H A370783 Rafael Jakimczuk, <a href="http://dx.doi.org/10.13140/RG.2.2.12174.13124">Arithmetical Functions over the Powerful Part of an Integer</a>, ResearchGate, 2024.
%F A370783 a(n) = A332880(A057521(n)).
%F A370783 Let f(n) = a(n)/A370784(n):
%F A370783 f(n) is multiplicative with f(p) = 1 and f(p^e) = 1 + 1/p for e >= 2.
%F A370783 f(n) = 1 if and only if n is squarefree (A005117).
%F A370783 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} f(k) = zeta(3)/zeta(6) = 1.181564... (A157289) (Jakimczuk, 2024).
%e A370783 Fractions begin with: 1, 1, 1, 3/2, 1, 1, 1, 3/2, 4/3, 1, 1, 3/2, ...
%t A370783 a[n_] := Numerator[Times @@ (1 + 1/Select[FactorInteger[n], Last[#] > 1 &][[;; , 1]])]; Array[a, 100]
%o A370783 (PARI) a(n) = {my(f = factor(n)); numerator(prod(i = 1, #f~, if(f[i,2] == 1, 1, 1 + 1/f[i,1])));}
%Y A370783 Cf. A005117, A057521, A157289, A295295, A332880, A370784 (denominators).
%K A370783 nonn,easy,frac
%O A370783 1,4
%A A370783 _Amiram Eldar_, Mar 02 2024