proposed

approved

editing

proposed

allocated for Clark KimberlingNumerator of least splitting rational of s(n) and s(n+1), where s(n) = 1 + 1/2^2 + ... + 1/n^2.

1, 4, 7, 10, 22, 3, 29, 20, 17, 14, 25, 36, 11, 30, 19, 46, 27, 35, 51, 91, 8, 141, 85, 61, 45, 82, 37, 95, 29, 50, 71, 113, 21, 97, 76, 55, 123, 34, 81, 47, 107, 60, 73, 86, 112, 138, 190, 307, 13, 395, 239, 174, 135, 109, 96, 83, 153, 70, 127, 57, 158, 101

1,2

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.

Clark Kimberling, <a href="/A227686/b227686.txt">Table of n, a(n) for n = 1..1000</a>

The denominators (A227685) and numerators (A227686) can be read from this chain: s(1) <= 1 < s(2) < 4/3 < s(3) < 7/5 < s(4) < 10/7 < s(5) < 22/15 < ...

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^(-2), {k, 1, n}]

t = Table[r[s[n], s[n + 1]], {n, 1, 150}] (*fractions)

fd = Denominator[t] (*A227685*)

fn = Numerator[t] (*A227686*)

Cf. A227631, A227685.

allocated

nonn,frac,easy

Clark Kimberling, Jul 19 2013

approved

editing

allocated for Clark Kimberling

allocated

approved