cp's OEIS Frontend

A366148 The sum of divisors of the cubefree part of n (A360539).

%I A366148 #8 Oct 02 2023 01:44:44
%S A366148 1,3,4,7,6,12,8,1,13,18,12,28,14,24,24,1,18,39,20,42,32,36,24,4,31,42,
%T A366148 1,56,30,72,32,1,48,54,48,91,38,60,56,6,42,96,44,84,78,72,48,4,57,93,
%U A366148 72,98,54,3,72,8,80,90,60,168,62,96,104,1,84,144,68,126,96
%N A366148 The sum of divisors of the cubefree part of n (A360539).
%H A366148 Amiram Eldar, <a href="/A366148/b366148.txt">Table of n, a(n) for n = 1..10000</a>
%F A366148 a(n) = A000203(A360539(n)).
%F A366148 a(n) = A000203(n)/A366146(n).
%F A366148 Multiplicative with a(p^e) = (p^(e+1)-1)/(p-1) if e <= 2, and 1 otherwise.
%F A366148 a(n) >= 1, with equality if and only if n is cubefull (A036966).
%F A366148 a(n) <= A000203(n), with equality if and only if n is cubefree (A004709).
%F A366148 Dirichlet g.f.: zeta(s) * Product_{p prime} (1 + 1/p^(s-1) + 1/p^(2*s-2) - 1/p^(3*s-2) - 1/p^(3*s-1)). CHECK
%F A366148 Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} (p/(p+1) + 1/p - 1/p^4) = 0.63884633697952950095... .
%t A366148 f[p_, e_] := If[e < 3, (p^(e+1)-1)/(p-1), 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A366148 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2] < 3, (f[i,1]^(f[i,2]+1)-1)/(f[i,1]-1), 1));}
%Y A366148 Cf. A000203, A004709, A036966, A092261, A360539, A366077, A366078, A366146, A366147.
%K A366148 nonn,easy,mult
%O A366148 1,2
%A A366148 _Amiram Eldar_, Oct 01 2023