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%I #7 Jul 25 2015 00:48:24
%S 1,3,4,6,7,9,10,12,14,15,17,18,20,21,23,24,26,28,29,31,32,34,35,37,39,
%T 40,42,43,45,46,48,49,51,53,54,56,57,59,60,62,64,65,67,68,70,71,73,74,
%U 76,78,79,81,82,84,85,87,89,90,92,93,95,96,98,99,101,103,104,106,107
%N Lower Beatty sequence for e^G, G = Euler's gamma constant.
%C The ratio of lower Beatty terms to upper tends to k = e^G. This can be confirmed by examining the continued fraction convergents to 1/k = 0.561459484..., the first few being 1/1, 1/2, 4/7, 5/9, 9/16, 32/57, ... Check: 32/57 = 0.562403508... Let a convergent = a/b. Through n = (a+b) = 14, 9 terms are in the lower Beatty pair set and 5 are in the upper (2, 5, 8, 11, 13).
%C Young, p. 245 states "It has been argued on probabilistic grounds that the expected number of primes p in the octave interval (x,2x) for which 2^p - 1 is a prime is e^G, where G is Euler's constant."
%D Robert M. Young, "Excursions in Calculus, An Interplay of the Continuous and the Discrete", MAA, p. 245.
%F a(n) = floor(n*(k+1)/k)). Lower Beatty pair terms are the set of natural numbers not in the set of upper Beatty pair terms (the latter in A093609).
%e a(7) = 10 = floor(10*(k+1)/k)), (k+1)/k = 1.56145948..., k = e^G = 1.78107241..., G = Euler's Gamma constant, 0.577215664...
%t Table[ Floor[n*(E^EulerGamma + 1)/(E^EulerGamma)], {n, 70}] (* _Robert G. Wilson v_, Apr 07 2004 *)
%Y Cf. A093609, A093608.
%K nonn
%O 1,2
%A _Gary W. Adamson_, Apr 04 2004
%E Corrected and extended by _Robert G. Wilson v_, Apr 07 2004
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