A227687 Least splitter of s(n) and s(n+1), where s(n) = 1/sqrt(1) + 1/sqrt(2) + ... + 1/sqrt(n).

1, 1, 2, 1, 2, 1, 3, 2, 1, 4, 2, 3, 1, 3, 2, 3, 1, 3, 2, 3, 1, 5, 3, 2, 3, 1, 5, 3, 2, 3, 5, 1, 4, 3, 2, 3, 5, 1, 4, 3, 2, 3, 4, 1, 6, 4, 3, 2, 3, 5, 1, 6, 4, 3, 2, 3, 4, 7, 1, 5, 3, 5, 2, 3, 4, 7, 1, 5, 4, 3, 2, 3, 4, 6, 1, 7, 4, 3, 5, 2, 3, 4, 6, 1, 8, 5

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.

The positions of 1 in this sequences (indicating those least splitting rationals of s(n) and s(n+1) which are integers) are given by A186351.

Links

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

Example

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 <  . . .

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*)

Crossrefs

Cf. A227631, A227688.

Sequence in context: A231568 A327525 A327540 * A128118 A205696 A029635

Adjacent sequences: A227684 A227685 A227686 * A227688 A227689 A227690

Keyword

nonn,frac,easy

Author

Clark Kimberling, Jul 21 2013

Status

approved