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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)).
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%I #29 Sep 03 2024 01:35:21

%S 1,3,3,18,39,71,323,536,1001,8544,45723,208524,398959,3400196,5394991,

%T 10391023,150869313,1097649283,5467464369,10622799089,132941053437,

%U 403978495031,403978495031,8286870547680,76601727404275,399178399621704,781379079653017

%N 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)).

%C 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).

%H Matthew House, <a href="/A227773/b227773.txt">Table of n, a(n) for n = 2..807</a>

%F f(n) = n!/!n = A000142(n)/A000166(n). - _Matthew House_, Aug 14 2024

%e The least splitters are the denominators of the least splitting rationals for f(n) and f(n+1):

%e 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

%e f(2) = 2 <= 2/1 < f(3) = 3;

%e f(3) = 3 > 8/3 >= f(4) = 8/3;

%e f(4) = 8/3 <= 8/3 < f(5) = 30/11;

%e f(5) = 30/11 > 49/18 >= f(6) = 144/53.

%t 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 *)

%Y Cf. A227631, A227688.

%K nonn,frac

%O 2,2

%A _Clark Kimberling_, Jul 30 2013

%E Corrected and edited by _Clark Kimberling_, Jun 26 2015

%E Corrected and extended by _Matthew House_, Aug 14 2024