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 A359451 #10 Mar 24 2025 22:35:55
%S A359451 2,0,8,6,3,7,7,6,6,5,0,0,5,9,8,8,7,1,6,0,8,9,7,5,5,8,5,6,7,3,4,1,3,2,
%T A359451 7,7,2,6,9,2,0,2,2,0,9,6,9,2,2,3,9,5,1,6,9,5,1,2,3,8,3,8,5,7,9,2,1,5,
%U A359451 3,2,0,0,0,2,8,2,1,0,0,2,6,1,4,7,1,6,0,5,8,4,8,5,2,6,7,0,9,4,9,0,7
%N A359451 Decimal expansion of Sum_{k>=1} 1/A359450(k).
%C A359451 The problem of calculating the sum of this series was proposed by David Smith in Bornemann et al. (2004). The value that is given here is from his solution on the web page of this book. He shows that the series is slowly converging. E.g., the sum of the first 2^2000 - 1 terms is 1.95403... .
%D A359451 Daniel D. Bonar and Michael J. Khoury, Jr., Real infinite Series, The Mathematical Association of America, 2006, pp. 159, 190-191.
%D A359451 Kiran S. Kedlaya, Daniel M. Kane, Jonathan M. Kane, and Evan M. O'Dorney, The William Lowell Putnam Mathematical Competition 2001-2016: Problems, Solutions, and Commentary, American Mathematical Society, 2020, pp. 86-87.
%H A359451 Folkmar Bornemann, Dirk Laurie, Stan Wagon, and Jörg Waldvogel, <a href="https://www-m3.ma.tum.de/m3old/bornemann/challengebook/">The SIAM 100-Digit Challenge, A Study in High-Accuracy Numerical Computing</a>, SIAM, Philadelphia, 2004. See <a href="https://www-m3.ma.tum.de/m3old/bornemann/challengebook/AppendixD/AppendixD.pdf">Appendix D</a>, Problem 2, p. 281.
%H A359451 David Smith, <a href="https://www-m3.ma.tum.de/m3old/bornemann/challengebook/AppendixD/">On a slowly converging sum</a>, Solution to Problem 2, Loyola Marymount University, Note of November 2003.
%F A359451 Equals 5/3 + Sum_{k>=3} (H(2^k-1)-H(2^(k-1)-1))/A359450(k), where H(k) = A001008(k)/A002805(k) is the k-th harmonic number.
%F A359451 Equals (1/(1-log(2))) * (5/3 - 3*log(2)/2 + Sum_{k>=3} (H(2^k-1)-H(2^(k-1)-1)-log(2))/A359450(k)).
%F A359451 Both formulas are from Smith (2003).
%e A359451 2.08637766500598871608975585673413277269202209692239...
%Y A359451 Cf. A001008, A002805, A359450.
%K A359451 nonn,cons,base
%O A359451 1,1
%A A359451 _Amiram Eldar_, Jan 02 2023