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

Offset: 0

Author

Alonso del Arte, Mar 22 2013

Keywords

• cons
• nonn
• easy

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

Examples

			0.810569469138702171551...
		

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.
• Index entries for transcendental numbers

Crossrefs

Cf. A008683, A059956, A092742, A111003 (reciprocal).

Programs

Mathematica

RealDigits[8/Pi^2, 10, 108][[1]]

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

Extensions

Mathematica program edited by Harvey P. Dale, Nov 17 2024