A227630 Numerator of the least splitting rational of the harmonic numbers H(n) and H(n+1).

1, 3, 2, 9, 7, 5, 8, 11, 17, 3, 31, 19, 13, 23, 10, 17, 31, 7, 25, 18, 11, 26, 15, 19, 23, 27, 39, 55, 107, 4, 73, 49, 37, 29, 25, 21, 38, 17, 47, 30, 56, 13, 48, 35, 22, 31, 40, 58, 112, 9, 68, 41, 32, 55, 23, 37, 51, 79, 14, 61, 47, 33, 71, 19, 62, 43, 24

Offset: 1

Author

Clark Kimberling, Jul 18 2013

Keywords

• nonn
• frac

Comments

See A227631 for the definition of least splitting rational.

Examples

			The first few splitting rationals are 1/1, 3/2, 2/1, 9/4, 7/3, 5/2, 8/3, 11/4, 17/6, 3/1, 31/10, 19/6; e.g. 9/4 splits H(4) and H(5), as indicated by H(4) = 1 + 1/2 + 1/3 + 1/4 =  2.083...  < 2.25 < 2.283... = H(5) and the chain H(1) <= 1/1 < H(2) < 3/2 < H(3) < 2/1 < H(4) < 9/4 < ...
		

Links

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

Crossrefs

Cf. A227629, 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[h[n], h[n + 1]], {n, 1, 120}];
Denominator[t] (* A227629 *)
Numerator[t]   (* A227630 *)  (* Peter J. C. Moses, Jul 15 2013 *)