Backhouse's constant is a mathematical constant named after Nigel Backhouse. Its value 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,
| infty | |
| P(x)=1+\sum | |
| k=1 |
pkxk=1+2x+3x2+5x3+7x4+ …
| Q(x)= | 1 |
| P(x) |
| infty | |
| =\sum | |
| k=0 |
qkxk.
\limk\left|
| qk+1 | |
| qk |
\right\vert=1.45607\ldots
This limit was conjectured to exist by Backhouse, and later proven by Philippe Flajolet.