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 A076393 #62 Sep 05 2025 04:47:51
%S A076393 1,2,6,4,0,8,4,7,3,5,3,0,5,3,0,1,1,1,3,0,7,9,5,9,9,5,8,4,1,6,4,6,6,9,
%T A076393 4,9,1,1,1,4,5,6,0,1,7,9,2,0,9,0,6,5,5,3,3,1,5,3,4,5,6,4,1,9,9,0,7,7,
%U A076393 5,9,0,1,6,3,6,2,9,5,1,6,0,1,4,2,2,6,3,9,0,9,2,6,8,3,9,8,5,1,5,0,4,8,0,3,3
%N A076393 Decimal expansion of Vardi constant arising in the Sylvester sequence.
%C A076393 Vardi showed A000058(n) = floor(c^(2^(n+1))+1/2) where c=1.26408473...
%C A076393 Named after the Canadian mathematician Ilan Vardi (b. 1957). - _Amiram Eldar_, Jun 22 2021
%C A076393 This constant is transcendental. - _Quanyu Tang_, Mar 20 2025
%D A076393 Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 443-448.
%D A076393 Ronald L. Graham, Donald E. Knuth and Oren Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 2nd. ed., 1994, exercise 4.37, p. 518.
%D A076393 Ilan Vardi, Computational Recreations in Mathematica, Addison-Wesley, 1991, pp. 82-89.
%H A076393 A. V. Aho and N. J. A. Sloane, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/11-4/aho-a.pdf">Some doubly exponential sequences</a>, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437.
%H A076393 A. V. Aho and N. J. A. Sloane, <a href="/A000058/a000058.pdf">Some doubly exponential sequences</a>, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437 (original plus references that F.Q. forgot to include - see last page!)
%H A076393 Thomas Bloom, <a href="https://www.erdosproblems.com/148">Let F(k) be the number of solutions to 1 = 1/n_1 + ... + 1/n_k, where 1 <= n_1 < ... < n_k are distinct integers. Find good estimates for F(k)</a>, Erdős Problems.
%H A076393 Matthew Brendan Crawford, <a href="http://hdl.handle.net/10919/90573">On the Number of Representations of One as the Sum of Unit Fractions</a>, Master's Thesis, Virginia Polytechnic Institute and State University (2019).
%H A076393 Artūras Dubickas, <a href="https://doi.org/10.1007/s11139-021-00428-5">Transcendency of some constants related to integer sequences of polynomial iterations</a>, Ramanujan J, Vol. 57, 2022, pp. 569-581.
%H A076393 Steven Finch, <a href="https://arxiv.org/abs/2411.16062">Exercises in Iterational Asymptotics</a>, arXiv:2411.16062 [math.NT], 2024. See p. 10.
%H A076393 Zheng Li and Quanyu Tang, <a href="https://arxiv.org/abs/2503.12277">On a conjecture of Erdős and Graham about the Sylvester's sequence</a>, arXiv:2503.12277 [math.NT], 2025. See p. 3.
%H A076393 Benjamin Nill, <a href="https://doi.org/10.1007/s00454-006-1299-y">Volume and lattice points of reflexive simplices</a>, Discrete & Computational Geometry, Vol. 37, No. 2 (2007), pp. 301-320; <a href="https://arxiv.org/abs/math/0412480">arXiv preprint</a>, arXiv:math/0412480 [math.AG], 2004-2007.
%H A076393 Terence Tao, <a href="https://github.com/teorth/erdosproblems/blob/main/README.md#table">Erdős problem database</a>, see no. 148.
%H A076393 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SylvestersSequence.html">Sylvester's Sequence</a>.
%F A076393 Equals lim_{n->oo} A000058(n)^(1/2^(n+1)). - _Robert FERREOL_, Feb 15 2019
%F A076393 Equals sqrt((3/2) * Product_{k>=0} (1 + 1/(2*A000058(k)-1)^2)^(1/2^(k+1))). - _Amiram Eldar_, Jun 22 2021
%e A076393 1.26408473530530111307959958416466949111456...
%t A076393 digits = 105; For[c = 2; olds = -1; s = 0; j = 1, RealDigits[olds, 10, digits+5] != RealDigits[s, 10, digits+5], j++; c = c^2-c+1, olds = s; s = s + 2^(-j-1)*Log[1+(2*c-1)^-2] // N[#, digits+5]&]; chi = Sqrt[6]/2*Exp[s]; RealDigits[chi, 10, digits] // First (* _Jean-François Alcover_, Jun 05 2014 *)
%Y A076393 Cf. A000058.
%K A076393 cons,easy,nonn,changed
%O A076393 1,2
%A A076393 _Benoit Cloitre_, Nov 06 2002