A227634 Least splitter of log(n) and log(n+1).

1, 1, 3, 2, 3, 5, 1, 6, 4, 3, 5, 2, 5, 3, 4, 5, 6, 10, 18, 1, 11, 8, 6, 5, 4, 7, 10, 3, 5, 7, 9, 15, 2, 11, 7, 5, 8, 14, 3, 10, 7, 4, 9, 5, 11, 6, 7, 8, 10, 12, 15, 21, 34, 1, 40, 24, 17, 13, 11, 10, 8, 7, 13, 6, 11, 5, 14, 9, 17, 4, 11, 7, 10, 13, 22, 3, 17

Offset: 1

Author

Clark Kimberling, Jul 18 2013

Keywords

• nonn
• frac
• easy

Comments

Essentially the same as A183163. - R. J. Mathar, Jul 27 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.

Examples

			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 < ...
		

Links

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

Crossrefs

Cf. A227631.

Programs

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 *)