A217739 - OEIS

A217739

Decimal expansion of 8/Pi^2.

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)/(1-1/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

The probability that the line that passes through two points selected independently and uniformly at random in the interior of a quadrant (quarter of a disk) intersects the arc at exactly one point (Zerr, 1891). - Amiram Eldar, Apr 13 2026

LINKS

Table of n, a(n) for n=0..96.

J. E. Nymann, On the probability that k positive integers are relatively prime II, Journal of Number Theory, Vol. 7, No. 4 (1975), pp. 406-412.

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

Eric Weisstein's World of Mathematics, Triangle Wave.

George B. McClellan Zerr, Solution to Problem 11135, Mathematical questions and solutions from the "Educational Times", Vol. 55 (1891), p. 162.

Index entries for transcendental numbers.

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} (1-1/k^2)^((-1)^k). - Amiram Eldar, Apr 09 2022

EXAMPLE

0.810569469138702171551...