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

Author

Reinhard Zumkeller, Dec 15 2003

Keywords

• nonn

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).

Examples

			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.
		

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

Crossrefs

Cf. A091050, A001597, A000203, A183104.

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

Programs

Mathematica

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

PARI

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

Formula

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