A070228 Number of perfect powers (A001597) not exceeding 2^n.

1, 1, 2, 3, 5, 8, 11, 16, 23, 31, 42, 58, 82, 114, 156, 217, 299, 417, 583, 814, 1136, 1589, 2224, 3116, 4369, 6136, 8623, 12128, 17064, 24023, 33839, 47689, 67227, 94805, 133738, 188710, 266351, 376019, 530941, 749820, 1059097, 1496144, 2113802, 2986770, 4220666

Offset: 0

Author

Donald S. McDonald, May 14 2002

Keywords

• nonn
• easy

Examples

			How many powers are there not exceeding 2^4?: 1, 4, 8, 9, 16. Hence a(4) = 5.
a(22)=2224: there are 2224 powers not exceeding 2^22.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 0..6643 (terms 0..1000 from Robert G. Wilson v)

Crossrefs

Cf. A070428, A188951.

Programs

Mathematica

f[n_] := 1 - Sum[ MoebiusMu[x]*Floor[2^(n/x) - 1], {x, 2, n}]; Array[f, 44, 0] (* Robert G. Wilson v, Jan 20 2015 *)

PARI

a(n) = 1 - sum(k=2, n, moebius(k)*(sqrtnint(2^n,k)-1));

Python

from sympy import mobius, integer_nthroot
def A070228(n): return int(1+sum(mobius(x)*(1-integer_nthroot(1<Chai Wah Wu, Aug 13 2024

Formula

a(n) = 1 - Sum_{k=2..n} Moebius(k)*floor(2^(n/k)-1). - Robert G. Wilson v, Jan 20 2015

a(n) = A188951(n) + 1 for n > 1. - Amiram Eldar, May 19 2022

Extensions

a(39)-a(44) from Alex Ratushnyak, Jan 02 2014