login
A227773
Least splitter of f(n) and f(n+1), where s(1) = 1, s(2) = 1, s(n) = s(n-1) + s(n-2)/(n-2) and f(n) = n/(n - s(n)).
1
1, 3, 3, 18, 39, 71, 323, 536, 1001, 8544, 45723, 208524, 398959, 3400196, 5394991, 10391023, 150869313, 1097649283, 5467464369, 10622799089, 132941053437, 403978495031, 403978495031, 8286870547680, 76601727404275, 399178399621704, 781379079653017
OFFSET
2,2
COMMENTS
Suppose that 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. Since f(n) -> e, the corresponding least splitting rationals (see Example) also approach e; e.g., |f(30) - e| < 10^(-33).
LINKS
FORMULA
f(n) = n!/!n = A000142(n)/A000166(n). - Matthew House, Aug 14 2024
EXAMPLE
The least splitters are the denominators of the least splitting rationals for f(n) and f(n+1):
2/1, 8/3, 8/3, 49/18, 106/39, 193/71, 878/323, 1457/536, 2721/1001, 23225/8544, 124288/45723, 566827/208524, 1084483/398959, 9242691/3400196, 14665106/5394991, 28245729/10391023
f(2) = 2 <= 2/1 < f(3) = 3;
f(3) = 3 > 8/3 >= f(4) = 8/3;
f(4) = 8/3 <= 8/3 < f(5) = 30/11;
f(5) = 30/11 > 49/18 >= f(6) = 144/53.
MATHEMATICA
z = 17; 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[1] = 1; s[2] = 1; s[n_] := s[n] = s[n - 1] + s[n - 2]/(n - 2); N[Table[s[k], {k, 1, z}]]; N[Table[k/(k - s[k]), {k, 2, z}], 20]; t = Table[r[n/(n - s[n]), (n + 1)/(n + 1 - s[n + 1])], {n, 2, z}]; fd = Denominator[t] (* Peter J. C. Moses, Jul 30 2013 *)
CROSSREFS
Sequence in context: A130184 A183039 A192527 * A346380 A371552 A189737
KEYWORD
nonn,frac
AUTHOR
Clark Kimberling, Jul 30 2013
EXTENSIONS
Corrected and edited by Clark Kimberling, Jun 26 2015
Corrected and extended by Matthew House, Aug 14 2024
STATUS
approved