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 power series such that the coefficients of successive terms are the prime numbers,

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

and its multiplicative inverse as a formal power series,

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

Then:

${\displaystyle \lim _{k\to \infty }\left|{\frac {q_{k+1}}{q_{k}}}\right\vert =1.45607\ldots }$.^[1]

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

References

1. Sloane, N. J. A. (ed.). "Sequence A072508". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
2. Backhouse, N. (1995). Formal reciprocal of a prime power series. unpublished note.
3. 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 Unpublished manuscript.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.

Further reading

• Weisstein, Eric W. "Backhouse's Constant". MathWorld.
• Sloane, N. J. A. (ed.). "Sequence A030018". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
• Sloane, N. J. A. (ed.). "Sequence A074269". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
• Sloane, N. J. A. (ed.). "Sequence A088751". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.