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Backhouse's Constant

download Mathematica trial version BackhousesConstant.nb

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

(1)
  (2)

where is taken to be 1. The function has a zero at (Sloane's A088751). Now let Q(x) be defined by
(3)
  (4)
  (5)

(Sloane's A030018).

Then N. Backhouse conjectured that


(Sloane's 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, ...] (Sloane's A074269), which is also the same as the continued fraction of except for a leading 0 in the latter.

Prime Number

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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." 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/.




cite this as

Eric W. Weisstein. "Backhouse's Constant." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/BackhousesConstant.html



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