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,

: $P(x)=1+\sum_{k=1}^\infty p_k x^k=1+2x+3x^2+5x^3+7x^4+\cdots$

and its multiplicative inverse as a formal power series,

: $Q(x)=\frac{1}{P(x)}=\sum_{k=0}^\infty q_k x^k.$

Then:

: $\lim_{k \to \infty}\left | \frac{q_{k+1}}{q_k} \right \vert = 1.45607\ldots$ .{{r|OEIS_A072508}}

This limit was conjectured to exist by Backhouse,{{r|Backhouse1995}} and later proven by Philippe Flajolet.{{r|Flajolet1995}}

References

{{reflist|refs=

{{cite book |last=Backhouse |first=N. |date=1995 |title=Formal reciprocal of a prime power series |series=unpublished note}}

{{cite book |last=Flajolet |first=Philippe |author-link=Philippe Flajolet |date=25 November 1995 |title=On the existence and the computation of Backhouse's constant |series=Unpublished manuscript}}
Reproduced in {{cite conference |last=Hwang |first=Hsien-Kuei |date=19 June 2014 |others=with Brigitte Vallée and Julien Clément |title=Les cahiers de Philippe Flajolet |conference=AofA 2014 - 25th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms |location=Paris |access-date=18 May 2021 |url=http://algo.stat.sinica.edu.tw/hk/?p=827 |conference-url=http://www.aofa14.upmc.fr/?sec=program }}

}}

Backhouse's constant

References

Further reading