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.
