This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A217739 #49 Feb 16 2025 08:33:18
%S A217739 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,
%T A217739 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,
%U A217739 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
%N A217739 Decimal expansion of 8/Pi^2.
%C A217739 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).
%C A217739 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.
%C A217739 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
%C A217739 The probability that two randomly chosen odd numbers will be coprime (Nymann, 1975). - _Amiram Eldar_, Aug 07 2020
%H A217739 J. E. Nymann, <a href="https://doi.org/10.1016/0022-314X(75)90044-X">On the probability that k positive integers are relatively prime II</a>, Journal of Number Theory, Vol. 7, No. 4 (1975), pp. 406-412.
%H A217739 Matt Springer, <a href="http://scienceblogs.com/builtonfacts/2009/08/16/sunday-function-43/">Sunday Function</a>, Built on Facts, Aug 16 2009, from ScienceBlogs.
%H A217739 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FourierSeriesTriangleWave.html">Triangle Wave</a>.
%H A217739 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>
%F A217739 Equals -Sum_{k>=1} mu(2*k)/k^2, where mu is the Möbius function (A008683). - _Amiram Eldar_, Aug 20 2020
%F A217739 Equals Product_{k>=2} (1-1/k^2)^((-1)^k). - _Amiram Eldar_, Apr 09 2022
%e A217739 0.810569469138702171551...
%t A217739 RealDigits[8/Pi^2, 10, 108][[1]]
%Y A217739 Cf. A008683, A059956, A092742, A111003 (reciprocal).
%K A217739 cons,nonn,easy
%O A217739 0,1
%A A217739 _Alonso del Arte_, Mar 22 2013
%E A217739 Mathematica program edited by _Harvey P. Dale_, Nov 17 2024