Backhouse's constant

Backhouse's constant is a mathematical constant named after Nigel Backhouse. Its value is approximately 1.456 074 948.

It is defined by using the

prime numbers

,

P(x)=1+\sum _{k=1}^{\infty }p_{k}x^{k}=1+2x+3x^{2}+5x^{3}+7x^{4}+\cdots

Q(x)={\frac {1}{P(x)}}=\sum _{k=0}^{\infty }q_{k}x^{k}.

Then:

\lim _{k\to \infty }\left|{\frac {q_{k+1}}{q_{k}}}\right\vert =1.45607\ldots

.^[1]

This limit was conjectured to exist by Backhouse,^[2] and later proven by Philippe Flajolet.^[3]

References

^ Sloane, N. J. A. (ed.). "Sequence A072508". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
^ Backhouse, N. (1995). Formal reciprocal of a prime power series. unpublished note.
^ Flajolet, Philippe (25 November 1995). On the existence and the computation of Backhouse's constant. Unpublished manuscript.
Reproduced in Hwang, Hsien-Kuei (19 June 2014). Les cahiers de Philippe Flajolet. AofA 2014 - 25th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms. with Brigitte Vallée and Julien Clément. Paris. Retrieved 18 May 2021.

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