A093610 Lower Beatty sequence for e^G, G = Euler's gamma constant.

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, 40, 42, 43, 45, 46, 48, 49, 51, 53, 54, 56, 57, 59, 60, 62, 64, 65, 67, 68, 70, 71, 73, 74, 76, 78, 79, 81, 82, 84, 85, 87, 89, 90, 92, 93, 95, 96, 98, 99, 101, 103, 104, 106, 107

Offset: 1

Author

Gary W. Adamson, Apr 04 2004

Keywords

• nonn

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."

Examples

			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...
		

References

• Robert M. Young, "Excursions in Calculus, An Interplay of the Continuous and the Discrete", MAA, p. 245.

Crossrefs

Cf. A093609, A093608.

Programs

Mathematica

Table[ Floor[n*(E^EulerGamma + 1)/(E^EulerGamma)], {n, 70}] (* Robert G. Wilson v, Apr 07 2004 *)

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).

Extensions

Corrected and extended by Robert G. Wilson v, Apr 07 2004