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.

REFERENCES:

Finch, S. R. "Kalmár's Composition Constant." §5.5 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 292-295, 2003.

Finch, S. "Kalmár's Composition Constant." https://algo.inria.fr/bsolve/.

Sloane, N. J. A. Sequences A030018, A072508, A074269, and A088751 in "The On-Line Encyclopedia of Integer Sequences."