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

Offset

0,2

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

Formula

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

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 *)

Prog

(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

Crossrefs

Cf. A005117, A013928.

Apart from initial term, same as A053462.

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

Sequence in context: A177132 A364430 A077642 * A259335 A127688 A111532

Adjacent sequences: A071169 A071170 A071171 * A071173 A071174 A071175

Keyword

nonn,changed

Author

Robert G. Wilson v, Jun 10 2002

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

Status

approved