A143658 Number of squarefree integers not exceeding 2^n.

1, 2, 3, 6, 11, 20, 39, 78, 157, 314, 624, 1245, 2491, 4982, 9962, 19920, 39844, 79688, 159360, 318725, 637461, 1274918, 2549834, 5099650, 10199301, 20398664, 40797327, 81594626, 163189197, 326378284, 652756722, 1305513583, 2611027094

Offset: 0

Author

M. F. Hasler, Aug 28 2008

Keywords

• nonn

Comments

Except for the first 2 terms, it would not make a difference to replace "not exceeding" by "less than": that sequence would start 0,1,3,6,11,20,39,78,...

Examples

			a(4) = 11 since there are the 11 squarefree integers {1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15} not exceeding 2^4=16.
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 0..73 (terms 0..58 from Gerard P. Michon, terms 59..64 from Peter Polm)
• Project Euler, Problem 193: Squarefree Numbers
• G. P. Michon, On the number of squarefree integers not exceeding N. - _Gerard P. Michon_, Apr 30 2009

Crossrefs

Cf. A005117, A013928, A071172, A053462.

Programs

Mathematica

c = 0; k = 1; lst = {1}; Do[ While[k <= 2^n, If[ SquareFreeQ@k, c++ ]; k++ ]; AppendTo[lst, c], {n, 27}] (* Robert G. Wilson v, Aug 31 2008 *)

PARI

print1(s=1);for(p=1,20,print1(", ",s+=sum(k=2^(p-1)+1, 2^p, issquarefree(k))))

PARI

a(n)=sum(d=1,sqrtint(n=2^n),moebius(d)*n\d^2) \\ Charles R Greathouse IV, Nov 14 2012

PARI

a(n)=my(s); forsquarefree(d=1,sqrtint(n=2^n), s += n\d[1]^2*moebius(d)); s \\ Charles R Greathouse IV, Jan 08 2018

Python

from math import isqrt
from sympy import mobius
def A143658(n):
    m = 1<Chai Wah Wu, Jun 01 2024

Formula

a(n) = Sum for i = 1 to 2^(n/2) of A008683(i)*floor(2^n/i^2). - Gerard P. Michon, Apr 30 2009

The limit of a(n)/2^n is 6/Pi^2. - Gerard P. Michon, Apr 30 2009

Extensions

5 more terms from Robert G. Wilson v, Aug 31 2008

More terms from Alexis Olson (AlexisOlson(AT)gmail.com), Nov 08 2008