|
Search: id:A093610
|
|
|
| A093610 |
|
Lower Beatty sequence for e^G, G = Euler's gamma constant. |
|
+0
3
|
|
| 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
(list; graph; listen)
|
|
|
OFFSET
|
1,2
|
|
|
COMMENT
|
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 = .561459484..., the first few being 1/1, 1/2, 4/7, 5/9, 9/16, 32/57...Check: 32/57 = .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, .577215664...
|
|
MATHEMATICA
|
Table[ Floor[n*(E^EulerGamma + 1)/(E^EulerGamma)], {n, 70}] (from Robert G. Wilson v Apr 07 2004)
|
|
CROSSREFS
|
Cf. A093609, A093608.
Sequence in context: A072561 A141206 A140100 this_sequence A140758 A094178 A085795
Adjacent sequences: A093607 A093608 A093609 this_sequence A093611 A093612 A093613
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 04 2004
|
|
EXTENSIONS
|
Corrected and extended by Robert G. Wilson v (rgwv(AT)rgwv.com), Apr 07 2004
|
|
|
Search completed in 0.002 seconds
|
