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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.
11
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, 12, 11
OFFSET
1,2
COMMENTS
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)?
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
EXAMPLE
Northwest corner:
m=1 m=2 m=3 m=4 m=5 m=6 m=7 m=8
n=1: 1, 4, 13, 40, 121, 364, 1093, 3280
n=2: 2, 4, 8, 16, 32, 64, 128, 256
n=3: 3, 5, 9, 16, 29, 53, 97, 178
n=4: 4, 6, 9, 14, 22, 35, 56, 90
n=5: 5, 7, 10, 15, 23, 36, 57, 91
n=6: 6, 8, 11, 16, 24, 36, 54, 81
n=7: 7, 9, 12, 16, 22, 31, 44, 63
n=8: 8, 10, 13, 17, 23, 32, 45, 64
The chain indicated by row n=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: A372493 A167557 A069836 * A125153 A191451 A193950
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Jul 21 2013
STATUS
approved