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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 June 28 03:21 EDT 2017. Contains 288813 sequences.