A076393 - OEIS

A076393

Decimal expansion of Vardi constant arising in the Sylvester sequence.

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, 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, 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

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OFFSET

1,2

COMMENTS

Vardi showed A000058(n) = floor(c^(2^(n+1))+1/2) where c=1.26408473...

Named after the Canadian mathematician Ilan Vardi (b. 1957). - Amiram Eldar, Jun 22 2021

This constant is transcendental. - Quanyu Tang, Mar 20 2025

REFERENCES

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 443-448.

Ronald L. Graham, Donald E. Knuth and Oren Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 2nd. ed., 1994, exercise 4.37, p. 518.

Ilan Vardi, Computational Recreations in Mathematica, Addison-Wesley, 1991, pp. 82-89.

LINKS

Table of n, a(n) for n=1..105.

A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437.

A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437 (original plus references that F.Q. forgot to include - see last page!)

Thomas Bloom, Problem 148 and Problem 315, Erdős Problems.

Matthew Brendan Crawford, On the Number of Representations of One as the Sum of Unit Fractions, Master's Thesis, Virginia Polytechnic Institute and State University (2019).

Artūras Dubickas, Transcendency of some constants related to integer sequences of polynomial iterations, Ramanujan J, Vol. 57, 2022, pp. 569-581.

Erdős problems database contributors, Erdős problem database, see no. 148.

Steven Finch, Exercises in Iterational Asymptotics, arXiv:2411.16062 [math.NT], 2024. See p. 10.

Zheng Li and Quanyu Tang, On a conjecture of Erdős and Graham about the Sylvester's sequence, arXiv:2503.12277 [math.NT], 2025. See p. 3.

Benjamin Nill, Volume and lattice points of reflexive simplices, Discrete & Computational Geometry, Vol. 37, No. 2 (2007), pp. 301-320; arXiv preprint, arXiv:math/0412480 [math.AG], 2004-2007.

Wouter van Doorn and Quanyu Tang, The smallest denominator not contained in a unit fraction decomposition of 1 with fixed length, arXiv:2512.22083 [math.NT], 2025. See p. 2.

Eric Weisstein's World of Mathematics, Sylvester's Sequence.

FORMULA

Equals lim_{n->oo} A000058(n)^(1/2^(n+1)). - Robert FERREOL, Feb 15 2019

Equals sqrt((3/2) * Product_{k>=0} (1 + 1/(2*A000058(k)-1)^2)^(1/2^(k+1))). - Amiram Eldar, Jun 22 2021

EXAMPLE

1.26408473530530111307959958416466949111456...

MATHEMATICA

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 *)

CROSSREFS

Cf. A000058.

Sequence in context: A332253 A379954 A247493 * A389021 A379953 A054674

Adjacent sequences: A076390 A076391 A076392 * A076394 A076395 A076396

KEYWORD

cons,easy,nonn

AUTHOR

Benoit Cloitre, Nov 06 2002

STATUS

approved