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

%I A227777 #38 Jul 15 2015 17:48:17
%S A227777 1,2,3,7,39,110,252,465,1001,9545,27634,136168,589394,398959,5394991,
%T A227777 36568060,130087267,312129649,5779594018,5467464369,69204258903,
%U A227777 186055048882,403978495031,8690849042711,25668568633102,246378923308185,1163579759684330
%N A227777 Least splitter of n-th and (n+1)st partial sums of 1/0! + 1/1! + ... + 1/n! + ... = e.
%C A227777 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.
%C A227777 Conjecture: a(n) <= n*sqrt(n!) for all n>0; see scatterplot under Links. - _Jon E. Schoenfield_, Jun 28 2015
%H A227777 Jon E. Schoenfield, <a href="/A227777/b227777.txt">Table of n, a(n) for n = 1..807</a>
%H A227777 Manfred Scheucher, <a href="/A227777/a227777.sage.txt">Sage Script</a>
%H A227777 Jon E. Schoenfield, <a href="/A227777/a227777.txt">Magma program</a>
%H A227777 Jon E. Schoenfield, <a href="/A227777/a227777_1.png">Scatterplot of a(n)/(n*sqrt(n!)) vs. n for n = 1..5000</a>
%e A227777 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).
%e A227777 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
%t A227777 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 *)
%Y A227777 Cf. A227631.
%K A227777 nonn,frac
%O A227777 1,2
%A A227777 _Clark Kimberling_, Jul 30 2013
%E A227777 a(16)-a(17) from _Manfred Scheucher_, Jun 23 2015
%E A227777 a(18)-a(19) from _Robert G. Wilson v_, Jun 27 2015
%E A227777 a(20)-a(27) from _Jon E. Schoenfield_, Jun 27 2015