A071172 Number of squarefree integers <= 10^n.

1, 7, 61, 608, 6083, 60794, 607926, 6079291, 60792694, 607927124, 6079270942, 60792710280, 607927102274, 6079271018294, 60792710185947, 607927101854103, 6079271018540405, 60792710185403794, 607927101854022750, 6079271018540280875, 60792710185402613302, 607927101854026645617

Offset: 0

Author

Robert G. Wilson v, Jun 10 2002

Keywords

• nonn

Comments

The limit of a(n)/10^n is 6/Pi^2 (see A059956). - Gerard P. Michon, Apr 30 2009

Links

• J. Pawlewicz, Table of n, a(n) for n = 0..36
• W. Hürlimann, A First Digit Theorem for Square-Free Integer Powers, Pure Mathematical Sciences, Vol. 3, 2014, no. 3, 129 - 139 HIKARI Ltd.
• G. P. Michon, On the number of squarefree integers not exceeding N. - _Gerard P. Michon_, Apr 30 2009
• J. Pawlewicz, Counting square-free numbers, arXiv preprint arXiv:1107.4890 [math.NT], 2011.
• Eric Weisstein's World of Mathematics, Squarefree

Crossrefs

Cf. A005117, A013928.

Apart from first two terms, same as A053462.

Binary counterpart is A143658. - Gerard P. Michon, Apr 30 2009

Programs

Mathematica

f[n_] := Sum[ MoebiusMu[i]Floor[n/i^2], {i, Sqrt@ n}]; Table[ f[10^n], {n, 0, 14}] (* Robert G. Wilson v, Aug 04 2012 *)

PARI

a(n)=sum(d=1,sqrtint(n=10^n),moebius(d)*n\d^2) \\ Charles R Greathouse IV, Nov 14 2012

PARI

a(n)=my(s); forsquarefree(d=1,sqrtint(n=10^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 A071172(n): return sum(mobius(k)*(10**n//k**2) for k in range(1,isqrt(10**n)+1)) # Chai Wah Wu, May 10 2024

Formula

a(n) = Sum_{i=1..10^(n/2)} A008683(i)*floor(10^n/i^2). - Gerard P. Michon, Apr 30 2009

Extensions

Extended by Eric W. Weisstein, Sep 14 2003

3 more terms from Jud McCranie, Sep 01 2005

4 more terms from Gerard P. Michon, Apr 30 2009