A091051 Sum of divisors of n that are perfect powers.

1, 1, 1, 5, 1, 1, 1, 13, 10, 1, 1, 5, 1, 1, 1, 29, 1, 10, 1, 5, 1, 1, 1, 13, 26, 1, 37, 5, 1, 1, 1, 61, 1, 1, 1, 50, 1, 1, 1, 13, 1, 1, 1, 5, 10, 1, 1, 29, 50, 26, 1, 5, 1, 37, 1, 13, 1, 1, 1, 5, 1, 1, 10, 125, 1, 1, 1, 5, 1, 1, 1, 58, 1, 1, 26, 5, 1, 1, 1, 29, 118, 1, 1, 5, 1, 1, 1, 13, 1, 10

Offset

1,4

Comments

a(n) = 1 iff n is squarefree: a(A005117(n))=1, a(A013929(n))>1;

a(p^k) = 1+(p^2)*(p^(k-1)-1)/(p-1) for p prime, k>0.

a(A000961(n)) = A086455(n)-A025473(n).

Links

Antti Karttunen, Table of n, a(n) for n = 1..16385

Eric Weisstein's World of Mathematics, Perfect Power

Eric Weisstein's World of Mathematics, Divisor Function

Index entries for sequences related to sums of divisors

Formula

G.f.: Sum_{k=i^j, i>=1, j>=2, excluding duplicates} k*x^k/(1 - x^k). - Ilya Gutkovskiy, Mar 20 2017

Example

Divisors of n=108: {1,2,3,4,6,9,12,18,27,36,54,108}, a(108) = 1^2 + 2^2 + 3^2 + 3^3 + 6^2 = 1+4+9+27+36 = 77.

Mathematica

a[n_] := DivisorSum[n, #*Boole[# == 1 || GCD @@ FactorInteger[#][[All, 2]] > 1]&]; Array[a, 90] (* Jean-François Alcover, May 09 2017 *)

Prog

(PARI) a(n) = sumdiv(n, d, d*((d==1) || ispower(d))); \\ Michel Marcus, Oct 02 2014

Crossrefs

Cf. A091050, A001597, A000203, A183104.

Differs from A183097 for the first time at n=72.

Sequence in context: A145295 A366990 A360722 * A183097 A353958 A285486

Adjacent sequences: A091048 A091049 A091050 * A091052 A091053 A091054

Keyword

nonn

Author

Reinhard Zumkeller, Dec 15 2003

Status

approved