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, 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

Author

Clark Kimberling, Jul 30 2013

Keywords

• nonn
• frac

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).

Examples

			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.
		

Links

• Matthew House, Table of n, a(n) for n = 2..807

Crossrefs

Cf. A227631, A227688.

Programs

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 *)

Formula

f(n) = n!/!n = A000142(n)/A000166(n). - Matthew House, Aug 14 2024

Extensions

Corrected and edited by Clark Kimberling, Jun 26 2015

Corrected and extended by Matthew House, Aug 14 2024