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A091050 Number of divisors of n that are perfect powers. 22
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

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);

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, Perfect Power

Eric Weisstein's World of Mathematics, Divisor Function

FORMULA

a(n) = A073093(n)-A001221(n) = A001222(n)-A001221(n)+1. - David W. Wilson, Aug 28 2007

a(n) = sum (A075802(A027750(n,k)): k=1..A000005(n)). - 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

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

CROSSREFS

Cf. A091051, A001597, A000005.

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

STATUS

approved

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Last modified February 17 23:35 EST 2020. Contains 332006 sequences. (Running on oeis4.)