A227688 -id:A227688 - cp's OEIS frontend

A227687 Least splitter of s(n) and s(n+1), where s(n) = 1/sqrt(1) + 1/sqrt(2) + ... + 1/sqrt(n).

1, 1, 2, 1, 2, 1, 3, 2, 1, 4, 2, 3, 1, 3, 2, 3, 1, 3, 2, 3, 1, 5, 3, 2, 3, 1, 5, 3, 2, 3, 5, 1, 4, 3, 2, 3, 5, 1, 4, 3, 2, 3, 4, 1, 6, 4, 3, 2, 3, 5, 1, 6, 4, 3, 2, 3, 4, 7, 1, 5, 3, 5, 2, 3, 4, 7, 1, 5, 4, 3, 2, 3, 4, 6, 1, 7, 4, 3, 5, 2, 3, 4, 6, 1, 8, 5

Offset: 1

Clark Kimberling, Jul 21 2013

The positions of 1 in this sequences (indicating those least splitting rationals of s(n) and s(n+1) which are integers) are given by A186351.

			The denominators (A227687) and numerators (A227688) can be read from these chains:
1 < 2 < 5/2 < 3 < 7/2 < 4 < 13/3 < 9/2 < 5 < 21/4 < 11/2 < 17/3 < 6 < . . .
s(1) <= 1 < s(2) < 2 < s(3) < 5/2 < s(4) < 3 < s(5) < 4 < s(6) < 13/3 <  . . .

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A227631, A227688.

r[x_, y_] := Module[{c, d}, d = NestWhile[#1 + 1 &, 1, ! (c = Ceiling[#1 x - 1]) < Ceiling[#1 y] - 1 &]; (c + 1)/d];
s[n_] := s[n] = Sum[k^(-1/2), {k, 1, n}]; t = Table[r[s[n], s[n + 1]], {n, 1, 15}] (*fractions*)
fd = Denominator[t] (*A227687*)
fn = Numerator[t]   (*A227688*)

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)).

Original entry on oeis.org

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

Clark Kimberling, Jul 30 2013

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).

			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.

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

Cf. A227631, A227688.

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 *)

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

Corrected and edited by Clark Kimberling, Jun 26 2015

Corrected and extended by Matthew House, Aug 14 2024

cp's OEIS Frontend

A227687 Least splitter of s(n) and s(n+1), where s(n) = 1/sqrt(1) + 1/sqrt(2) + ... + 1/sqrt(n).

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