

A093610


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


2



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
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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



