OFFSET
1,2
COMMENTS
Suppose x < y. The least splitter of x and y is introduced at A227631 as the least positive integer d such that x <= c/d < y for some integer c; the number c/d is called the least splitting rational of x and y. Let s(n) = 1/0! + 1/1! + ... + 1/n!; since s(n) -> e, the corresponding least splitting rationals (see Example) also approach e.
Conjecture: a(n) <= n*sqrt(n!) for all n>0; see scatterplot under Links. - Jon E. Schoenfield, Jun 28 2015
LINKS
Jon E. Schoenfield, Table of n, a(n) for n = 1..807
Manfred Scheucher, Sage Script
Jon E. Schoenfield, Magma program
Jon E. Schoenfield, Scatterplot of a(n)/(n*sqrt(n!)) vs. n for n = 1..5000
EXAMPLE
The first 19 splitting rationals are 2, 5/2, 8/3, 19/7, 106/39, 299/110, 685/252, 1264/465, 2721/1001, 25946/9545, 75117/27634, 370143/136168, 1602139/589394, 1084483/398959, 14665106/5394991, 99402293/36568060, 353613854/130087267, 848456353/312129649 & 15710565395/5779594018. Regarding the last one, |15710565395/5779594018 - e| < 10^(-19).
The numerators of these rationals are a proper subsequence of A006258 & A119014 and the denominators are a proper subsequence of A006259 & A119015. - Robert G. Wilson v, Jun 27 2015
MATHEMATICA
z = 16; r[x_, y_] := Module[{a, b, x1 = Min[{x, y}], y1 = Max[{x, y}]}, If[x == y, x, b = NestWhile[#1 + 1 &, 1, ! (a = Ceiling[#1 x1 - 1]) < Ceiling[#1 y1] - 1 &]; (a + 1)/b]]; s[n_] := s[n] = Sum[1/(k - 1)!, {k, 1, n}]; N[Table[s[k], {k, 1, z}]]; t = Table[r[s[n], s[n + 1]], {n, 2, z}]; fd = Denominator[t] (* Peter J. C. Moses, Jul 20 2013 *)
CROSSREFS
KEYWORD
nonn,frac
AUTHOR
Clark Kimberling, Jul 30 2013
EXTENSIONS
a(16)-a(17) from Manfred Scheucher, Jun 23 2015
a(18)-a(19) from Robert G. Wilson v, Jun 27 2015
a(20)-a(27) from Jon E. Schoenfield, Jun 27 2015
STATUS
approved