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A091051 Sum of divisors of n that are perfect powers.

%I A091051 #26 Feb 16 2025 08:32:52
%S A091051 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,
%T A091051 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,
%U A091051 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
%N A091051 Sum of divisors of n that are perfect powers.
%C A091051 a(n) = 1 iff n is squarefree: a(A005117(n))=1, a(A013929(n))>1;
%C A091051 a(p^k) = 1+(p^2)*(p^(k-1)-1)/(p-1) for p prime, k>0.
%C A091051 a(A000961(n)) = A086455(n)-A025473(n).
%H A091051 Antti Karttunen, <a href="/A091051/b091051.txt">Table of n, a(n) for n = 1..16385</a>
%H A091051 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PerfectPower.html">Perfect Power</a>
%H A091051 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DivisorFunction.html">Divisor Function</a>
%H A091051 <a href="/index/Su#sums_of_divisors">Index entries for sequences related to sums of divisors</a>
%F A091051 G.f.: Sum_{k=i^j, i>=1, j>=2, excluding duplicates} k*x^k/(1 - x^k). - _Ilya Gutkovskiy_, Mar 20 2017
%e A091051 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.
%t A091051 a[n_] := DivisorSum[n, #*Boole[# == 1 || GCD @@ FactorInteger[#][[All, 2]] > 1]&]; Array[a, 90] (* _Jean-François Alcover_, May 09 2017 *)
%o A091051 (PARI) a(n) = sumdiv(n, d, d*((d==1) || ispower(d))); \\ _Michel Marcus_, Oct 02 2014
%Y A091051 Cf. A091050, A001597, A000203, A183104.
%Y A091051 Differs from A183097 for the first time at n=72.
%K A091051 nonn
%O A091051 1,4
%A A091051 _Reinhard Zumkeller_, Dec 15 2003