Backhouse's Constant


Let P(x) be defined as the power series whose nth term has a coefficient equal to the nth prime p_n,


The function has a zero at x_0=-0.68677... (OEIS A088751). Now let Q(x) be defined by


(OEIS A030018).


Then N. Backhouse conjectured that


(OEIS A072508). This limit was subsequently shown to exist by P. Flajolet. Note that B=-1/x_0, 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 -x_0 except for a leading 0 in the latter.

See also

Prime Number

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

Referenced on Wolfram|Alpha

Backhouse's Constant

Cite this as:

Weisstein, Eric W. "Backhouse's Constant." From MathWorld--A Wolfram Web Resource.

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