Backhouse's constant

Binary	1.01110100110000010101001111101100…
Decimal	1.45607494858268967139959535111654…
Hexadecimal	1.74C153ECB002353B12A0E476D3ADD…
Continued fraction	${\displaystyle 1+{\cfrac {1}{2+{\cfrac {1}{5+{\cfrac {1}{5+{\cfrac {1}{4+\ddots }}}}}}}}}$

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:

${\displaystyle \lim _{k\to \infty }\left|{\frac {q_{k+1}}{q_{k}}}\right\vert =1.45607\ldots }$ (sequence A072508 in the OEIS).

This limit was conjectured to exist by Backhouse (1995), and the conjecture was later proven by Philippe Flajolet (1995).

References

• Backhouse, N. (1995), Formal reciprocal of a prime power series, unpublished note
• Flajolet, Philippe (November 25, 1995), On the existence and the computation of Backhouse's constant, Unpublished manuscript. Reproduced in Les cahiers de Philippe Flajolet, Hsien-Kuei Hwang, June 19, 2014, accessed 2014-12-06.
• Weisstein, Eric W. "Backhouse's Constant". MathWorld.

•  A030018,  A074269,  A088751