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
LINKS
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.
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...
MATHEMATICA
RealDigits[8/Pi^2, 10, 108]
CROSSREFS
KEYWORD
AUTHOR
Alonso del Arte, Mar 22 2013
STATUS
approved