Backhouse's constant

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

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

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• Backhouse, N. (1995), Formal reciprocal of a prime power series, unpublished note
• Flajolet, Philippe (November 25, 1995), On the existence and the computation of Backhouse's constant, Unpublished manuscript.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.
• 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.