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

0,2

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.

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

Example

There are 608 squarefree integers smaller than 1000.

Mathematica

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

Prog

(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

Crossrefs

Cf. A005117, A013928.

Cf. A059956, A063035.

Apart from initial term, same as A071172.

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

Sequence in context: A069618 A155458 A152281 * A160751 A236670 A142970

Adjacent sequences: A053459 A053460 A053461 * A053463 A053464 A053465

Keyword

nonn

Author

Harvey P. Dale, Aug 01 2001

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

Status

approved