Backhouse,s Constant
المؤلف:
Sloane, N. J. A.
المصدر:
Sequences A030018, A072508, A074269, and A088751 in "The On-Line Encyclopedia of Integer Sequences."
الجزء والصفحة:
...
1-10-2020
1061
Backhouse's Constant
Let 
be defined as the power series whose 
th term has a coefficient equal to the 
th prime 
,
The function has a zero at 
(OEIS A088751). Now let 
be defined by
(OEIS A030018).
Then N. Backhouse conjectured that
(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."
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