A129871 A variant of Sylvester's sequence: a(0)=1 and for n>0, a(n) = (a(0)*a(1)*...*a(n-1)) + 1.

1, 2, 3, 7, 43, 1807, 3263443, 10650056950807, 113423713055421844361000443, 12864938683278671740537145998360961546653259485195807

Offset: 0

Author

Ben Branman, Sep 16 2011

Keywords

• nonn

Comments

A variant of A000058, starting with an extra 1.

References

• Jean-Marie Monier, Analyse, Exercices corrigés, 2ème année, MP, Dunod, 1997, Exercice 3.3.4 page 284.

Links

• Mohammad K. Azarian, Sylvester's Sequence and the Infinite Egyptian Fraction Decomposition of 1, Problem 958, College Mathematics Journal, Vol. 42, No. 4, September 2011, p. 330.
• Mohammad K. Azarian, Sylvester's Sequence and the Infinite Egyptian Fraction Decomposition of 1, Solution College Mathematics Journal, Vol. 43, No. 4, September 2012, pp. 340-342.
• Junnosuke Koizumi, Irrationality of the reciprocal sum of doubly exponential sequences, arXiv:2504.05933 [math.NT], 2025.
• Vjekoslav Kovač, On simultaneous rationality of two Ahmes series, arXiv:2406.17593 [math.NT], 2024.

Crossrefs

Cf. A000058 which is the main entry for this sequence.

Cf. A118227.

Programs

Haskell

a129871 n = a129871_list !! n
a129871_list = 1 : a000058_list  -- Reinhard Zumkeller, Dec 18 2013

Mathematica

a[0] = 1; a[n_] := a[n] = Product[a[k], {k, 0, n - 1}] + 1

Formula

For n>0, a(n) = A000058(n-1).

a(1) = 2, a(n+1) = a(n)^2 - a(n) + 1. a(n) = round(c^(2^n)), where c = 1.264... is the Vardi constant, A076393. - Thomas Ordowski, Jun 11 2013

From Bernard Schott, Apr 06 2021: (Start)

Sum_{n>=0} 1/a(n) = 2.

Sum_{n>=0} (-1)^(n+1)/a(n) = 2 * (A118227 - 1). (End)

Extensions

Corrected and rewritten by Ben Branman, Sep 16 2011

Edited by Max Alekseyev, Oct 11 2012