Backhouse's Constant

Let $P(x)$ be defined as the Power series whose $n$th term has a Coefficient equal to the $n$th Prime,

$$P(x)\equiv \sum_{k=0}^\infty p_k x^k=1+2x+3x^2+5x^3+7x^4+11x^5+\ldots,$$

and let $Q(x)$ be defined by

$$Q(x)={1\over P(x)}=\sum_{k=0}^\infty q_k x^k.$$

Then N. Backhouse conjectured that

$$\lim_{n\to\infty}\left\vert{q_{n+1}\over q_n}\right\vert=1.456074948582689671399595351116\ldots.$$

The constant was subsequently shown to exist by P. Flajolet.

References

Finch, S. ``Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/backhous/backhous.html