Let 
be defined as the power series whose 
th term has a coefficient equal to the 
th prime 
,
 |  |  | (1) |
 |  |  | (2) |
The function has a zero at 
(OEIS A088751). Now let 
be defined by
 |  |  | (3) |
 |  |  | (4) |
 |  |  | (5) |
(OEIS A030018).
Then N. Backhouse conjectured that
 |  |  | (6) |
 |  |  | (7) |
(OEIS A072508). This limit was subsequently shown to exist by P. Flajolet. Note that 
, which follows from the radius of convergence of the reciprocal power series.
The continued fraction of Backhouse's constant is [1, 2, 5, 5, 4, 1, 1, 18, 1, 1, 1, 1, 1, 2, ...] (OEIS A074269), which is also the same as the continued fraction of 
except for a leading 0 in the latter.
