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A091050 Number of divisors of n that are perfect powers. 22

%I

%S 1,1,1,2,1,1,1,3,2,1,1,2,1,1,1,4,1,2,1,2,1,1,1,3,2,1,3,2,1,1,1,5,1,1,

%T 1,4,1,1,1,3,1,1,1,2,2,1,1,4,2,2,1,2,1,3,1,3,1,1,1,2,1,1,2,6,1,1,1,2,

%U 1,1,1,5,1,1,2,2,1,1,1,4,4,1,1,2,1,1,1,3,1,2,1,2,1,1,1,5,1,2,2,4,1,1

%N Number of divisors of n that are perfect powers.

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

%C a(p^k)=k for p prime, k>0: a(A000961(n))=A025474(n);

%C not the same as A005361: a(72)=5 <> A005361(72)=6.

%H Reinhard Zumkeller, <a href="/A091050/b091050.txt">Table of n, a(n) for n = 1..10000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PerfectPower.html">Perfect Power</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DivisorFunction.html">Divisor Function</a>

%F a(n) = A073093(n)-A001221(n) = A001222(n)-A001221(n)+1. - _David W. Wilson_, Aug 28 2007

%F a(n) = sum (A075802(A027750(n,k)): k=1..A000005(n)). - _Reinhard Zumkeller_, Dec 13 2012

%F G.f.: Sum_{k=i^j, i>=1, j>=2, excluding duplicates} x^k/(1 - x^k). - _Ilya Gutkovskiy_, Mar 20 2017

%e Divisors of n=108: {1,2,3,4,6,9,12,18,27,36,54,108},

%e a(108) = #{1^2, 2^2, 3^2, 3^3, 6^2} = 5.

%t ppQ[n_] := GCD @@ Last /@ FactorInteger@ n > 1; ppQ[1] = True; f[n_] := Length@ Select[ Divisors@ n, ppQ]; Array[f, 105] (* _Robert G. Wilson v_, Dec 12 2012 *)

%o (Haskell)

%o a091050 = sum . map a075802 . a027750_row

%o -- _Reinhard Zumkeller_, Dec 13 2012

%o (PARI) a(n) = 1+ sumdiv(n, d, ispower(d)>1); \\ _Michel Marcus_, Sep 21 2014

%Y Cf. A091051, A001597, A000005.

%K nonn

%O 1,4

%A _Reinhard Zumkeller_, Dec 15 2003

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Last modified April 1 05:29 EDT 2020. Contains 333155 sequences. (Running on oeis4.)