A119015
Denominators of "Farey fraction" approximations to e.
Original entry on oeis.org
0, 1, 1, 1, 1, 2, 3, 4, 7, 11, 18, 25, 32, 39, 71, 110, 181, 252, 323, 394, 465, 536, 1001, 1537, 2538, 3539, 4540, 5541, 6542, 7543, 8544, 9545, 18089, 27634, 45723, 63812, 81901, 99990, 118079, 136168, 154257, 172346, 190435, 208524, 398959, 607483
Offset: 0
The fractions are 1/0, 0/1, 1/1, 2/1, 3/1, 5/2, 8/3, 11/4, 19/7, ...
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f[x_, n_] := (m = Floor[x]; f0 = {m, m+1/2, m+1}; r = ({a___, b_, c_, d___} /; b < x < c) :> {b, (Numerator[b] + Numerator[c]) / (Denominator[b] + Denominator[c]), c};
Join[{m, m+1}, NestList[# /. r &, f0, n-3][[All, 2]]]);
Join[{0, 1, 1}, f[E, 43] // Denominator]
(* Jean-François Alcover, May 18 2011 *)
A006258
Numerators of approximations to e.
Original entry on oeis.org
1, 2, 3, 5, 8, 11, 19, 30, 49, 68, 87, 106, 193, 299, 492, 685, 878, 1071, 1264, 1457, 2721, 4178, 6899, 9620, 12341, 15062, 17783, 20504, 23225, 25946, 49171, 75117, 124288, 173459, 222630, 271801, 320972, 370143, 419314, 468485, 517656, 566827, 1084483, 1651310, 2735793
Offset: 1
A006258/A006259 = 1, 2, 3, 5/2, 8/3, 11/4, 19/7, 30/11, 49/18, 68/25, 87/32, 106/39, 193/71, 299/110, 492/181, ... .
- R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 122.
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
More terms from Matthew White (white_dawg_70(AT)hotmail.com), May 05 2003
A227777
Least splitter of n-th and (n+1)st partial sums of 1/0! + 1/1! + ... + 1/n! + ... = e.
Original entry on oeis.org
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
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
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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 *)
Showing 1-3 of 3 results.
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