Backhouse's constant

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Backhouse's constant

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Binary	1.01110100110000010101001111101100…
Decimal	1.45607494858268967139959535111654…
Hexadecimal	1.74C153ECB002353B12A0E476D3ADD…
Continued fraction	$1 + \cfrac{1}{2 + \cfrac{1}{5 + \cfrac{1}{5 + \cfrac{1}{4 + \ddots}}}}$ Note that this continued fraction is not periodic.

Backhouse's constant is a mathematical constant founded by N. Backhouse and 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:

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

and where

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

Then:

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

The limit was conjectured to exist by Backhouse which was later proved by P. Flajolet.

References

Weisstein, Eric W., "Backhouse's Constant" from MathWorld.
A030018, A074269, A088751

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