Backhouse's Constant

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.