A053462 Number of positive squarefree integers less than 10^n.

0, 6, 61, 608, 6083, 60794, 607926, 6079291, 60792694, 607927124, 6079270942, 60792710280, 607927102274, 6079271018294, 60792710185947, 607927101854103, 6079271018540405, 60792710185403794, 607927101854022750, 6079271018540280875, 60792710185402613302

Offset: 0

Author

Harvey P. Dale, Aug 01 2001

Keywords

• nonn

Examples

			There are 608 squarefree integers smaller than 1000.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 0..36 (from the b-file at A071172; terms 0..20 from Charles R Greathouse IV)
• Gérard P. Michon, On the number of squarefree integers not exceeding N.

Crossrefs

Cf. A005117, A013928.

Cf. A059956, A063035.

Apart from first two terms, same as A071172.

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

Programs

Mathematica

a[n_] := Module[{t=10^n-1}, Sum[MoebiusMu[k]Floor[t/k^2], {k, 1, Sqrt[t]}]]

PARI

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

PARI

a(n)=my(s); forsquarefree(d=1,sqrtint(n=10^n-1), 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 A053462(n):
    m = 10**n-1
    return sum(mobius(k)*(m//k**2) for k in range(1, isqrt(m)+1)) # Chai Wah Wu, Jun 01 2024

Formula

a(n)/10^n = (6/Pi^2)*(1+o(1)), cf. A059956.

a(n) = A071172(n) - [n <= 1] where [] is the Iverson bracket. - Chai Wah Wu, Jun 01 2024

Extensions

More terms from Dean Hickerson and Vladeta Jovovic, Aug 06 2001

One more term from Jud McCranie, Sep 01 2005

a(0)=0 and a(14)-a(17) from Gerard P. Michon, Apr 30 2009

a(18)-a(20) from Charles R Greathouse IV, Jan 08 2018