Backhouse's constant

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:

: $lim\_\{k\; o\; infty\}left\; |\; frac\{q\_\{k+1\{q\_k\}\; ight\; vert\; =\; 1.45607ldots$ OEIS|id=A072508.

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

References

*mathworld|title=Backhouse's Constant|urlname=BackhousesConstant
*OEIS2C|id=A030018, OEIS2C|id=A074269, OEIS2C|id=A088751