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 A382497 #16 May 12 2025 00:28:50 %S A382497 7,1,0,3,2,0,5,3,3,4,1,3,7,0,0,1,7,2,7,5,0,5,7,7,3,4,2,2,8,1,0,3,0,8, %T A382497 4,9,8,5,2,4,7,8,9,9,9,1,7,8,7,1,8,0,8,3,3,7,8,1,3,9,9,7,1,7,9,7,3,1, %U A382497 3,5,8,9,5,2,1,4,6,4,6,1,0,5,9,9,6,4,2,2,1,1 %N A382497 Decimal expansion of 3*log(x0)/(log(8*x0/3) - 8 + Pi/sqrt(3)), where x0 is the unique real root of 96*x^3 - 786663*x^2 + 17288*x - 96 = 0. %C A382497 It is proved that the irrationality measure of Pi is at most this value. Note that for N1 and N3 defined on page 12 in the article of Zeilberger and Zudilin are given by |N1| = 8/(3*sqrt(x0)) and N3 = 8*x0/3. %H A382497 Li Lai, Johannes Sprang, and Wadim Zudilin, <a href="https://arxiv.org/abs/2505.05005">A note on the irrationality of zeta_2(5)</a>, arXiv:2505.05005 [math.NT], 2025. See p. 2. %H A382497 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/IrrationalityMeasure.html">Irrationality Measure</a>. %H A382497 Wikipedia, <a href="https://en.wikipedia.org/wiki/Irrationality_measure">Irrationality measure</a>. %H A382497 Doron Zeilberger and Wadim Zudilin, <a href="https://arxiv.org/abs/1912.06345">The irrationality measure of π is at most 7.103205334137...</a>, arXiv:1912.06345 [math.NT], 2019-2020; Moscow Journal of Combinatorics and Number Theory, 9 (4): 407-419. %e A382497 x0 = 8194.38427358233563630075... %e A382497 mu(Pi) <= 3*log(x0)/(log(8*x0/3) - 8 + Pi/sqrt(3)) = 7.10320533413700172750... %o A382497 (PARI) my(x0 = solve(x=8194, 8195, 96*x^3 - 786663*x^2 + 17288*x - 96)); 3*log(x0)/(log(8*x0/3) - 8 + Pi/sqrt(3)) %Y A382497 Cf. A000796, A002194, A093602. %K A382497 nonn,cons %O A382497 1,1 %A A382497 _Jianing Song_, Mar 29 2025