Let P(x) be defined as the power series whose nth term has a coefficient equal to the nth prime
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where is taken to be 1. The function has a zero at (Sloane's A088751). Now let Q(x) be defined by
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(Sloane's A030018).
Then N. Backhouse conjectured that
(Sloane's A072508). This limit was subsequently shown to exist by P. Flajolet. Note that
The continued fraction of Backhouse's constant is [1, 2, 5, 5, 4, 1, 1, 18, 1, 1, 1, 1, 1, 2, ...] (Sloane's A074269), which is also the same as the continued fraction of except for a leading 0 in the latter.
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." http://pauillac.inria.fr/algo/bsolve/.
Sloane, N. J. A. Sequences A030018, A072508, A074269, and A088751 in "The On-Line Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/sequences/.
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