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A224820 Array r(n,m), where r(n,1) = n; r(n,2) = least k such that H(k) - H(n) > 1/n; and for m > 2, r(n,m) = least k such that H(k)-H(r(n,m-1)) > H(r(n,m-1)) - H(r(n,m-2)), where H = harmonic number.

%I A224820 #23 Feb 22 2025 21:30:36
%S A224820 1,4,2,13,4,3,40,8,5,4,121,16,9,6,5,364,32,16,9,7,6,1093,64,29,14,10,
%T A224820 8,7,3280,128,53,22,15,11,9,8,9841,256,97,35,23,16,12,10,9,29524,512,
%U A224820 178,56,36,24,16,13,11,10,88573,1024,327,90,57,36,22,17,14,12,11
%N A224820 Array r(n,m), where r(n,1) = n; r(n,2) = least k such that H(k) - H(n) > 1/n; and for m > 2, r(n,m) = least k such that H(k)-H(r(n,m-1)) > H(r(n,m-1)) - H(r(n,m-2)), where H = harmonic number.
%C A224820 For every n, the sequence H(r(n,m)) - H(r(n,m-1)) converges as m -> oo. Which row-sequences are linearly recurrent? Is r(4,m) = 1 + F(m+3), where F = A000045 (Fibonacci numbers)?
%C A224820 More generally, suppose that x and y are positive integers and that x <= y. Let c(1) = y and c(2) = least k such that H(k) - H(y) > H(y) - H(x); for n > 2, let c(n) = least k such that H(k) - H(c(n-1)) > H(c(n-1)) - H(c(n-2)). Thus the Egyptian fractions for m >= x are partitioned, and 1/x + ... + 1/c(1) < 1/(c(1)+1) + ... + 1/(c(2)) < 1/(c(2)+1) + ... + 1/(c(3)) < ... The sequences H(c(n))-H(c(n-1)) and c(n)/c(n-1) converge. For what choices of (x,y) is the sequence c(n) linearly recurrent?
%H A224820 Clark Kimberling, <a href="/A224820/b224820.txt">Table of n, a(n) for n = 1..1830</a>
%e A224820 Northwest corner:
%e A224820        m=1  m=2  m=3  m=4  m=5  m=6   m=7   m=8
%e A224820   n=1:   1,   4,  13,  40, 121, 364, 1093, 3280
%e A224820   n=2:   2,   4,   8,  16,  32,  64,  128,  256
%e A224820   n=3:   3,   5,   9,  16,  29,  53,   97,  178
%e A224820   n=4:   4,   6,   9,  14,  22,  35,   56,   90
%e A224820   n=5:   5,   7,  10,  15,  23,  36,   57,   91
%e A224820   n=6:   6,   8,  11,  16,  24,  36,   54,   81
%e A224820   n=7:   7,   9,  12,  16,  22,  31,   44,   63
%e A224820   n=8:   8,  10,  13,  17,  23,  32,   45,   64
%e A224820 The chain indicated by row n=4 is
%e A224820 1/4 < 1/5 + 1/6 < 1/7 + 1/8 + 1/9 < 1/10 + ... + 1/14 < ...
%t A224820 h[n_] := h[n] = HarmonicNumber[N[n, 300]]; z = 12; Table[s = 0; a[1] = NestWhile[# + 1 &, x + 1, ! (s += 1/#) >= h[x] - h[x - 1] &];   s = 0; a[2] = NestWhile[# + 1 &, a[1] + 1, ! (s += 1/#) >= h[a[1]] - h[x] &]; Do[test = h[a[t - 1]] - h[a[t - 2]] + h[a[t - 1]]; s = 0; a[t] = Floor[x /. FindRoot[h[x] == test, {x, a[t - 1]}, WorkingPrecision -> 100]] + 1, {t, 3, z}]; Flatten[{x, Map[a, Range[z]]}], {x, 1, z}] // TableForm (* A224820 array *)
%t A224820 t = Flatten[Table[%[[n - k + 1]][[k]], {n, z}, {k, n, 1, -1}]]; (* A224820 sequence *) (* _Peter J. C. Moses_, Jul 20 2013 *)
%Y A224820 Cf. A225918.
%K A224820 nonn,tabl
%O A224820 1,2
%A A224820 _Clark Kimberling_, Jul 21 2013