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

1,2

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.

Links

Table of n, a(n) for n=1..69.

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

Cf. A093609, A093608.

Sequence in context: A140100 A292642 A249117 * A206912 A330113 A329923

Adjacent sequences: A093607 A093608 A093609 * A093611 A093612 A093613

Keyword

nonn

Author

Gary W. Adamson, Apr 04 2004

Extensions

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

Status

approved