Backhouse's constant

Backhouse's constant is a mathematical constant named after Nigel Backhouse. Its value is approximately 1.456074948.

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]