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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.

%I A227631 #19 Sep 03 2024 01:35:02
%S A227631 1,3,2,10,6,5,30,18,7,4,82,50,15,8,16,226,136,21,13,20,9,615,372,42,
%T A227631 23,24,12,14,1673,1014,59,38,36,25,19,44,4549,2758,115,64,45,35,22,56,
%U A227631 17,12366,7500,161,106,55,70,26,73,33,11,33616,20389,315,175,67
%N 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.
%C A227631 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?
%C A227631 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
%H A227631 Matthew House, <a href="/A227631/b227631.txt">Table of n, a(n) for n = 1..10011</a> (rows 1..141).
%e A227631 Northwest corner of the array:
%e A227631   1 ... 3 ... 10 ... 30 ... 82 ... 226
%e A227631   2 ... 6 ... 18 ... 50 ... 136 .. 372
%e A227631   5 ... 7 ... 15 ... 21 ... 42 ... 59
%e A227631   4 ... 8 ... 13 ... 23 ... 38 ... 64
%e A227631   16 .. 20 .. 24 ... 36 ... 45 ... 55
%e A227631   9 ... 12 .. 25 ... 35 ... 70 ... 97
%e A227631   14 .. 19 .. 22 ... 26 ... 34 ... 40
%e A227631 t(2,1) = 2 matches 1 + 1/2 <= 3/2 < 1 + 1/2 + 1/3;
%e A227631 similarly, t(2,2) = 6 matches H(6) <= 5/2 < H(7) and t(2,3) = 18 matches H(18) <= 7/2 < H(19).
%t A227631 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}];
%t A227631 d = Denominator[t]; u[n_] := Flatten[Position[d, n]]; TableForm[Table[u[n], {n, 1, 50}]]  (* A227631  *)
%t 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 *)
%Y A227631 Cf. A227629, A227630.
%K A227631 nonn,tabl,frac
%O A227631 1,2
%A A227631 _Clark Kimberling_, Jul 18 2013