A072292 Number of proper powers b^d <= n (b > 1, d > 1).

0, 0, 0, 1, 1, 1, 1, 2, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 6, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11

Offset: 1

Author

Reinhard Zumkeller, Jul 12 2002

Keywords

• nonn
• easy
• nice

Comments

Base b = 1 is excluded since 1 would be 1^d for any degree d (degree of power not well defined).

Links

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

Crossrefs

a(i)=A069637(i) for i<36=6^2. Cf. A001597.

Cf. A075802 (first differences).

Programs

Haskell

a072292 n = a072292_list !! (n-1)
a072292_list = scanl (+) 0 $ tail a075802_list
-- Reinhard Zumkeller, May 26 2012

Mathematica

a[n_] := (pp = Reap[ Do[ If[b^d <= n, Sow[b^d]], {b, 2, Sqrt[n]}, {d, 2, Log[2, n]}]]; If[pp == {Null, {}}, 0, Length[ Union[ pp[[2, 1]]]]]); Table[a[n], {n, 1, 90}](* Jean-François Alcover, May 16 2012 *)
Module[{nn=10,pp},pp=Union[Flatten[Table[a^b,{a,2,nn},{b,2,nn}]]];Accumulate[ Table[ If[ MemberQ[pp,n],1,0],{n,2^nn}]]] (* Harvey P. Dale, Nov 14 2022 *)

PARI

A072292(n)=n=floor(n)+.5;-sum(k=2,log(n)\log(2),floor(n^(1/k)-1)*moebius(k))
\\ Charles R Greathouse IV, Sep 07 2010

Python

from sympy import mobius, integer_nthroot
def A072292(n): return int(-sum(mobius(k)*(integer_nthroot(n,k)[0]-1) for k in range(2,n.bit_length()))) # Chai Wah Wu, Mar 11 2025

Extensions

Edited by Daniel Forgues, Mar 03 2009