OFFSET
1,2
COMMENTS
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).
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."
REFERENCES
Robert M. Young, "Excursions in Calculus, An Interplay of the Continuous and the Discrete", MAA, p. 245.
FORMULA
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).
EXAMPLE
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...
MATHEMATICA
Table[ Floor[n*(E^EulerGamma + 1)/(E^EulerGamma)], {n, 70}] (* Robert G. Wilson v, Apr 07 2004 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Gary W. Adamson, Apr 04 2004
EXTENSIONS
Corrected and extended by Robert G. Wilson v, Apr 07 2004
STATUS
approved