

A217739


Decimal expansion of 8/Pi^2.


7



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

0,1


COMMENTS

This is the probability that a randomly chosen singly even number is squarefree. (The probability that any randomly chosen integer is squarefree is 6/Pi^2).
This number also arises in the study of the Fourier series for a triangle wave. By Equation 6 given by Weisstein, this number is b_1, since b_n = 8/(Pi^2 n^2) for odd n. Springer labels this a_1.
This is also the probability that the greatest common divisor of two randomly chosen positive integers will be a power of 2. Generally, the probability that the greatest common divisor of two random integers will be a power of p, a prime, is (6/Pi^2)/(11/p^2). Here we are considering the integer 1 to be a power of p.  Geoffrey Critzer, Jan 13 2015
The probability that two randomly chosen odd numbers will be coprime (Nymann, 1975).  Amiram Eldar, Aug 07 2020


LINKS

Matt Springer, Sunday Function, Built on Facts, Aug 16 2009, from ScienceBlogs.


FORMULA

Equals Sum_{k>=1} mu(2*k)/k^2, where mu is the Möbius function (A008683).  Amiram Eldar, Aug 20 2020
Equals Product_{k>=2} (11/k^2)^((1)^k).  Amiram Eldar, Apr 09 2022


EXAMPLE

0.810569469138702171551...


MATHEMATICA

RealDigits[8/Pi^2, 10, 108]


CROSSREFS



KEYWORD



AUTHOR



STATUS

approved



