A227629 Least splitter of the harmonic numbers H(n) and H(n+1).

1, 2, 1, 4, 3, 2, 3, 4, 6, 1, 10, 6, 4, 7, 3, 5, 9, 2, 7, 5, 3, 7, 4, 5, 6, 7, 10, 14, 27, 1, 18, 12, 9, 7, 6, 5, 9, 4, 11, 7, 13, 3, 11, 8, 5, 7, 9, 13, 25, 2, 15, 9, 7, 12, 5, 8, 11, 17, 3, 13, 10, 7, 15, 4, 13, 9, 5, 11, 17, 6, 7, 15, 8, 9, 11, 12, 14, 17

Offset

1,2

Comments

See A227631 for the definition of least splitter.

Links

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

Example

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 < ...

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 *)

Crossrefs

Cf. A227630, A227631.

Sequence in context: A336280 A362806 A277749 * A183201 A082467 A106407

Adjacent sequences: A227626 A227627 A227628 * A227630 A227631 A227632

Keyword

nonn,frac

Author

Clark Kimberling, Jul 18 2013

Status

approved