A227684 Numerator of least splitting rational of log(n) and log(n+1).

0, 1, 4, 3, 5, 9, 2, 13, 9, 7, 12, 5, 13, 8, 11, 14, 17, 29, 53, 3, 34, 25, 19, 16, 13, 23, 33, 10, 17, 24, 31, 52, 7, 39, 25, 18, 29, 51, 11, 37, 26, 15, 34, 19, 42, 23, 27, 31, 39, 47, 59, 83, 135, 4, 161, 97, 69, 53, 45, 41, 33, 29, 54, 25, 46, 21, 59, 38

Offset

1,3

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.

Links

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

Example

The splitting rationals of consecutive numbers log(1), log(2), ... are 0, 1, 4/3, 3/2, 5/3, 9/5, 2, 13/6, 9/4, 7/3, 12/5, 5/2, 13/5; the denominators form A227634, and the numerators, A227684.  Chain:
log(1) <= 0 < log(2) < 1 < log(3) < 4/3 < log(4) < 3/2 < log(5) < 5/3 < ...

Mathematica

h[n_] := h[n] = HarmonicNumber[n]; r[x_, y_] := Module[{c, d}, d = NestWhile[#1 + 1 &, 1, ! (c = Ceiling[#1 x - 1]) < Ceiling[#1 y] - 1 &]; (c + 1)/d]; t = Table[r[Log[n], Log[n + 1]], {n, 1, 120}] (*fractions*)
Denominator[t] (* A227634 *)
Numerator[t]  (* A227684 *)

Crossrefs

Cf. A227631, A227634.

Sequence in context: A256367 A242910 A200350 * A200636 A229938 A226654

Adjacent sequences: A227681 A227682 A227683 * A227685 A227686 A227687

Keyword

nonn,frac,easy

Author

Clark Kimberling, Jul 19 2013

Status

approved