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A227631
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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.
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16
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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
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1,2
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COMMENTS
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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?
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LINKS
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EXAMPLE
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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).
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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