A227777 Least splitter of n-th and (n+1)st partial sums of 1/0! + 1/1! + ... + 1/n! + ... = e.

1, 2, 3, 7, 39, 110, 252, 465, 1001, 9545, 27634, 136168, 589394, 398959, 5394991, 36568060, 130087267, 312129649, 5779594018, 5467464369, 69204258903, 186055048882, 403978495031, 8690849042711, 25668568633102, 246378923308185, 1163579759684330

Offset: 1

Author

Clark Kimberling, Jul 30 2013

Keywords

• nonn
• frac

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

Examples

			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
		

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

Crossrefs

Cf. A227631.

Programs

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 *)

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