A227688 Numerator of least splitting rational of s(n) and s(n+1), where s(n) = 1/sqrt(1) + 1/sqrt(2) + ... + 1/sqrt(n).

1, 2, 5, 3, 7, 4, 13, 9, 5, 21, 11, 17, 6, 19, 13, 20, 7, 22, 15, 23, 8, 41, 25, 17, 26, 9, 46, 28, 19, 29, 49, 10, 41, 31, 21, 32, 54, 11, 45, 34, 23, 35, 47, 12, 73, 49, 37, 25, 38, 64, 13, 79, 53, 40, 27, 41, 55, 97, 14, 71, 43, 72, 29, 44, 59, 104, 15

Offset: 1

Author

Clark Kimberling, Jul 21 2013

Keywords

• nonn
• frac
• easy

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.

Examples

			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 <  . . .
		

Links

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

Crossrefs

Cf. A227631, A227687.

Programs

Mathematica

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*)