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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(n,r(m-1)) - H(r(n,m-2), where H = harmonic number. 10
 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, 8, 7, 3280, 128, 53, 22, 15, 11, 9, 8, 9841, 256, 97, 35, 23, 16, 12, 10, 9, 29524, 512, 178, 56, 36, 24, 16, 13, 11, 10, 88573, 1024, 327, 90, 57, 36, 22, 17, 14 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS For every n, the sequence H(n,r(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)? 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? LINKS Clark Kimberling, Table of n, a(n) for n = 1..1830 EXAMPLE Northwest corner: 1...4...13..20...121..364..1093..9841 2...4...8...16...32...64...128...256 3...5...9...16...29...53...97....178 4...6...9...14...22...35...56....90 5...7...10..15...23...36...57....91 6...8...11..16...24...36...54....81 7...9...12..16...22...31...44....63 8...10..13..17...23...32...45....64 The chain indicated by row 4 is 1/4 < 1/5 + 1/6 < 1/7 + 1/8 + 1/9 < 1/10 + ... + 1/14 < ... MATHEMATICA 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 = Flatten[Table[%[[n - k + 1]][[k]], {n, z}, {k, n, 1, -1}]]; (* A224820 sequence *) (* Peter J. C. Moses, Jul 20 2013 *) CROSSREFS Cf. A225918. Sequence in context: A105196 A167557 A069836 * A125153 A191451 A193950 Adjacent sequences:  A224817 A224818 A224819 * A224821 A224822 A224823 KEYWORD nonn,tabl AUTHOR Clark Kimberling, Jul 21 2013 STATUS approved

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Last modified May 20 02:22 EDT 2019. Contains 323411 sequences. (Running on oeis4.)