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

1,4

Comments

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

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.

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

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} = 5.

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 *)

Prog

(Haskell)
a091050 = sum . map a075802 . a027750_row
-- Reinhard Zumkeller, Dec 13 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

Crossrefs

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

Sequence in context: A072411 A290107 A212180 * A005361 A303915 A322885

Adjacent sequences: A091047 A091048 A091049 * A091051 A091052 A091053

Keyword

nonn

Author

Reinhard Zumkeller, Dec 15 2003

Extensions

Wrong formula deleted by Amiram Eldar, Apr 29 2020

Status

approved