

A227631


Array t(n,k): row n consists of the positive integers m for which the least splitter of H(m) and H(m+1) is n, where H denotes harmonic number.


16



1, 3, 2, 10, 6, 5, 30, 18, 7, 4, 82, 50, 15, 8, 16, 226, 136, 21, 13, 20, 9, 615, 372, 42, 23, 24, 12, 14, 1673, 1014, 59, 38, 36, 25, 19, 44, 4549, 2758, 115, 64, 45, 35, 22, 56, 17, 12366, 7500, 161, 106, 55, 70, 26, 73, 33, 11, 33616, 20389, 315, 175, 67
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OFFSET

1,2


COMMENTS

Suppose that x < y. The least splitter of x and y is introduced here 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. Conjecture: every row of the array in A227631 is infinite, and every positive integer occurs exactly once. Let r be the limiting ratio of consecutive terms of row 1; is r = e?


LINKS

Table of n, a(n) for n=1..60.


EXAMPLE

Northwest corner of the array:
1 ... 3 ... 10 ... 30 ... 82 ... 226
2 ... 6 ... 18 ... 50 ... 136 .. 372
5 ... 7 ... 15 ... 21 ... 42 ... 59
4 ... 8 ... 13 ... 23 ... 38 ... 64
16 .. 20 .. 24 ... 36 ... 45 ... 55
9 ... 12 .. 25 ... 35 ... 70 ... 97
14 .. 19 .. 22 ... 26 ... 34 ... 40
t(2,1) = 2 matches 1 + 1/2 <= 3/2 < 1 + 1/2 + 1/3;
similarly, t(2,2) = 6 matches H(6) < 5/2 < H(7) and t(2,3) = 18 matches H(18) < 7/2 < H(19).


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, 40000}];
d = Denominator[t]; u[n_] := Flatten[Position[d, n]]; TableForm[Table[u[n], {n, 1, 50}]] (* A227631 *)
r1[n_, k_] := u[n][[k]]; z = 11; v = Flatten[Table[r1[n  k + 1, k], {n, z}, {k, n, 1, 1}]] (* A227631 sequence *) (* Peter J. C. Moses, Jul 15 2013 *)


CROSSREFS

Cf. A227629, A227630.
Sequence in context: A057977 A063549 A071653 * A246830 A268531 A056861
Adjacent sequences: A227628 A227629 A227630 * A227632 A227633 A227634


KEYWORD

nonn,tabl,frac


AUTHOR

Clark Kimberling, Jul 18 2013


STATUS

approved



