A091050 Number of divisors of n that are perfect powers.

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, 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, 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

Offset: 1

Author

Reinhard Zumkeller, Dec 15 2003

Keywords

• nonn

Comments

Not the same as A005361: a(72)=5 <> A005361(72)=6.

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} = 5.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Divisor Function.
• Eric Weisstein's World of Mathematics, Perfect Power.

Crossrefs

Cf. A000005, A001597, A005361, A072102, A091051.

Programs

Haskell

a091050 = sum . map a075802 . a027750_row
-- Reinhard Zumkeller, Dec 13 2012

Mathematica

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 *)

PARI

a(n) = 1+ sumdiv(n, d, ispower(d)>1); \\ Michel Marcus, Sep 21 2014

PARI

a(n)={my(f=factor(n)[,2]); 1 + if(#f, sum(k=2, vecmax(f), moebius(k)*(1 - prod(i=1, #f, 1 + f[i]\k))))} \\ Andrew Howroyd, Aug 30 2020

Formula

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

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

a(n) = Sum_{k=1..A000005(n)} A075802(A027750(n,k)). - Reinhard Zumkeller, Dec 13 2012

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

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1 + A072102 = 1.874464... . - Amiram Eldar, Dec 31 2023

Extensions

Wrong formula deleted by Amiram Eldar, Apr 29 2020