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.

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

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?

For any reduced fraction c/d with c sufficiently large, there exists an (H(m),H(m+1)) pair bracketing it to within 1/LCM(1,2,3,...,d), so d is the least splitter for all such pairs, and every row is infinite. Since each (H(m),H(m+1)) pair is assigned to a single row, each positive integer m occurs exactly once by construction. Since t(1,k) = A002387(k) - 1 for all k >= 1, r = e is indeed the limiting ratio for row 1. - Matthew House, Aug 14 2024

Links

Matthew House, Table of n, a(n) for n = 1..10011 (rows 1..141).

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