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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,

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

and let $Q(x)$ be defined by

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

Then N. Backhouse conjectured that

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

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




© 1996-9 Eric W. Weisstein
1999-05-26