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A227777
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Least splitter of n-th and (n+1)st partial sums of 1/0! + 1/1! + ... + 1/n! + ... = e.
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1
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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
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OFFSET
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1,2
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COMMENTS
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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
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LINKS
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EXAMPLE
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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).
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn,frac
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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