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:

\( \lim_{k \to \infty}\left | \frac{q_{k+1}}{q_k} \right \vert = 1.45607\ldots \) (sequence A072508 in OEIS).

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

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

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Backhouse, N. (1995), Formal reciprocal of a prime power series, unpublished note
Flajolet, Philippe (November 25, 1995), On the existence and the computation of Backhouse's constant, Unpublished manuscript.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.
OEIS A030018, OEIS A074269, OEIS A088751

Mathematics Encyclopedia