Backhouse's constant

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


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