A063035 -id:A063035 - cp's OEIS frontend

A229099 Decimal expansion of 1 - 6/Pi^2.

3, 9, 2, 0, 7, 2, 8, 9, 8, 1, 4, 5, 9, 7, 3, 3, 7, 1, 3, 3, 6, 7, 2, 3, 2, 2, 0, 7, 4, 1, 6, 3, 4, 1, 6, 6, 5, 7, 3, 8, 4, 7, 3, 5, 1, 9, 6, 6, 5, 2, 0, 7, 0, 6, 9, 2, 6, 3, 4, 5, 8, 0, 8, 6, 3, 4, 9, 6, 1, 2, 7, 4, 2, 2, 6, 5, 8, 7, 3, 5, 2, 8, 5, 2, 7, 4, 4, 3, 5, 6, 4, 4

Offset: 0

Charles R Greathouse IV, Sep 13 2013

Probability that a random number is not squarefree; probability that two random numbers have a common divisor greater than 1.

			0.39207289814597337133672322074163416657384735196652070692634580863496...

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000
Index entries for transcendental numbers

Cf. A000796, A002388, A013661, A059956, A063035.

RealDigits[1-1/Zeta[2],10,90][[1]] (* Stefano Spezia, Feb 21 2025 *)

PARI
```
1-6/Pi^2
```

Equals 1 - 1/zeta(2). - Stefano Spezia, Feb 21 2025

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

Original entry on oeis.org

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

Offset: 0

Harvey P. Dale, Aug 01 2001

nonn

			There are 608 squarefree integers smaller than 1000.

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.

Cf. A005117, A013928.
Cf. A059956, A063035.
Apart from first two terms, same as A071172.
Binary counterpart is A143658. - Gerard P. Michon, Apr 30 2009

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

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

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

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

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

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

A124580 Where A124579 has two successive identical values.

Original entry on oeis.org

1, 9, 15, 31, 36, 40, 47, 165, 237, 330, 354, 357, 365, 402, 406, 421, 426, 794, 797, 813, 885, 894, 897, 905, 914, 1257, 1281, 1290, 1298, 1301, 1337, 1522, 1526, 1545, 1842, 1865, 2094, 2098, 2118, 2121, 2137, 2569, 3598, 4602, 4609, 4621, 4629, 4726, 4729

Offset: 1

Robert G. Wilson v, Nov 05 2006, Nov 12 2006

nonn

Except for a(2), a(5) & a(6) none of these duplicates involve zeros.

			Interval 10^n . # of -1 ...# of 0 . # of 1 # of terms
............0.........0.........0.........1........1
............1.........4.........3.........3........1
............2........30........39........31........5
............3.......303.......392.......305.......18
............4......3053......3917......3030.......57
............5.....30421.....39206.....30373......189
............6....303857....392074....304069......636
............7...3039127...3920709...3040164.....1176
............8..30395383..39207306..30397311.....4621
............9.303963673.392072876.303963451....15952

Robert G. Wilson v, Table of n, a(n) for n=1..9066

Cf. A124579, Column 2 = A063035, Column 1 + Column 3 = A053462.

p = q = y = z = a = 0; s = {}; Do[ q = Switch[ MoebiusMu@n, -1, y++, 0, z++, 1, a++ ]; If[p == q, AppendTo[s, n - 1]]; p = q, {n, 5000}]; s

cp's OEIS Frontend

A229099 Decimal expansion of 1 - 6/Pi^2.

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

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A124580 Where A124579 has two successive identical values.

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Mathematica