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A327564 If n = Product (p_j^k_j) then a(n) = Product ((p_j + 1)^(k_j - 1)).

%I A327564 #26 Dec 07 2023 01:42:43
%S A327564 1,1,1,3,1,1,1,9,4,1,1,3,1,1,1,27,1,4,1,3,1,1,1,9,6,1,16,3,1,1,1,81,1,
%T A327564 1,1,12,1,1,1,9,1,1,1,3,4,1,1,27,8,6,1,3,1,16,1,9,1,1,1,3,1,1,4,243,1,
%U A327564 1,1,3,1,1,1,36,1,1,6,3,1,1,1,27,64,1,1,3,1
%N A327564 If n = Product (p_j^k_j) then a(n) = Product ((p_j + 1)^(k_j - 1)).
%H A327564 Antti Karttunen, <a href="/A327564/b327564.txt">Table of n, a(n) for n = 1..16383</a>
%H A327564 Antti Karttunen, <a href="/A327564/a327564.txt">Data supplement: n, a(n) computed for n = 1..65537</a>
%H A327564 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>.
%H A327564 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>.
%F A327564 a(1) = 1; a(n) = -Sum_{d|n, d<n} (-1)^A001221(n/d) * A003557(n/d) * a(d).
%F A327564 a(n) = A003959(n) / A048250(n) = A003968(n) / A007947(n).
%F A327564 a(n) = A348038(n) * A348039(n) = A340368(n) / A173557(n). - _Antti Karttunen_, Oct 29 2021
%F A327564 Dirichlet g.f.: 1/(zeta(s-1) * Product_{p prime} (1 - 1/p^(s-1) - 1/p^s)). - _Amiram Eldar_, Dec 07 2023
%e A327564 a(12) = a(2^2 * 3) = (2 + 1)^(2 - 1) * (3 + 1)^(1 - 1) = 3.
%t A327564 a[n_] := Times @@ ((#[[1]] + 1)^(#[[2]] - 1) & /@ FactorInteger[n]); Table[a[n], {n, 1, 85}]
%o A327564 (PARI) a(n) = my(f=factor(n)); for (k=1, #f~, f[k,1]++; f[k,2]--); factorback(f); \\ _Michel Marcus_, Mar 03 2020
%Y A327564 Cf. A001221, A003557, A003958, A003959, A003968, A005117 (positions of 1's), A007947, A048250, A064478, A064549, A173557, A323363, A325126, A326297, A340368, A348038, A348039, A347960.
%K A327564 nonn,mult
%O A327564 1,4
%A A327564 _Ilya Gutkovskiy_, Mar 03 2020