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

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, 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, 1, 1, 3, 1, 1, 1, 36, 1, 1, 6, 3, 1, 1, 1, 27, 64, 1, 1, 3, 1

Offset

1,4

Links

Antti Karttunen, Table of n, a(n) for n = 1..16383

Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Index entries for sequences computed from exponents in factorization of n.

Index entries for sequences computed from indices in prime factorization.

Formula

a(1) = 1; a(n) = -Sum_{d|n, d<n} (-1)^A001221(n/d) * A003557(n/d) * a(d).

a(n) = A003959(n) / A048250(n) = A003968(n) / A007947(n).

a(n) = A348038(n) * A348039(n) = A340368(n) / A173557(n). - Antti Karttunen, Oct 29 2021

Dirichlet g.f.: 1/(zeta(s-1) * Product_{p prime} (1 - 1/p^(s-1) - 1/p^s)). - Amiram Eldar, Dec 07 2023

Example

a(12) = a(2^2 * 3) = (2 + 1)^(2 - 1) * (3 + 1)^(1 - 1) = 3.

Mathematica

a[n_] := Times @@ ((#[[1]] + 1)^(#[[2]] - 1) & /@ FactorInteger[n]); Table[a[n], {n, 1, 85}]

Prog

(PARI) a(n) = my(f=factor(n)); for (k=1, #f~, f[k, 1]++; f[k, 2]--); factorback(f); \\ Michel Marcus, Mar 03 2020

Crossrefs

Cf. A001221, A003557, A003958, A003959, A003968, A005117 (positions of 1's), A007947, A048250, A064478, A064549, A173557, A323363, A325126, A326297, A340368, A348038, A348039, A347960.

Sequence in context: A365427 A366787 A348038 * A243748 A340149 A340075

Adjacent sequences: A327561 A327562 A327563 * A327565 A327566 A327567

Keyword

nonn,mult

Author

Ilya Gutkovskiy, Mar 03 2020

Status

approved