A070428 Number of perfect powers (A001597) not exceeding 10^n.

1, 4, 13, 41, 125, 367, 1111, 3395, 10491, 32670, 102231, 320990, 1010196, 3184138, 10046921, 31723592, 100216745, 316694005, 1001003332, 3164437425, 10004650118, 31632790244, 100021566157, 316274216762, 1000100055684

Offset: 0

Author

Donald S. McDonald, May 15 2002

Keywords

• easy
• nonn

Comments

In the programs for this sequence, 4*n can be replaced by the smaller floor(n*log(10)/log(2)). - T. D. Noe, Nov 17 2006

Examples

			a(1) = 4 because the powers 1, 4, 8, 9 do not exceed 10^1.
a(2) = 13 because 1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81 & 100, are the only perfect power numbers less than or equal to 100.
		

References

• The Dominion (Wellington, NZ), 'wtd sell', 9 Nov. 1991.
• sci.math, powers not exceeding n. nz science monthly advt, March 1993, 1:80 integers 1..10000 is perfect square or higher power.

Links

• Chai Wah Wu, Table of n, a(n) for n = 0..1999 (terms 0..999 from Robert G. Wilson v)
• Eric Weisstein's World of Mathematics, Perfect Power

Crossrefs

Cf. A001597.

Cf. A089579, A089580 (number of perfect powers (not including 1) < 10^n).

Programs

Mathematica

f[n_] := 1 - Sum[ MoebiusMu[x]*Floor[10^(n/x) - 1], {x, 2, n*Log2[10]}]; Array[f, 25, 0] (* Robert G. Wilson v, May 22 2009; modified Aug 04 2014 *)

PARI

for(n=0, 25, print1(sum(x=2, 4*n,-moebius(x)*(floor(10^(n/x)-1)))+1, ", ")); \\ Slightly modified by Jinyuan Wang, Mar 02 2020

Python

from sympy import mobius, integer_nthroot
def A070428(n): return int(1-sum(mobius(x)*(integer_nthroot(10**n,x)[0]-1) for x in range(2,(10**n).bit_length()))) # Chai Wah Wu, Aug 13 2024

Formula

a(n) ~ sqrt(10^n).

Extensions

More terms from Larry Reeves (larryr(AT)acm.org), Oct 03 2002

Edited and extended by Robert G. Wilson v, Oct 11 2002