

A227965


a(1) = least k such that 1 + 1/2 < H(k)  H(2); a(2) = least k such that H(a(1))  1/2 < H(k) H(a(1)), and for n > 2, a(n) = least k such that H(a(n1))  H(a(n2) > H(k)  H(a(n1)), where H = harmonic number.


5



11, 53, 249, 1164, 5435, 25371, 118428, 552798, 2580343, 12044484, 56221045, 262427666, 1224955522, 5717827134, 26689578960, 124581175389, 581517950673, 2714399875409, 12670230858892, 59141894115145, 276061555506087, 1288595564424512, 6014885070144844
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OFFSET

1,1


COMMENTS

Suppose that x and y are positive integers and that x <=y. Let a(1) = least k such that H(y)  H(x1) < H(k)  H(y); let a(2) = least k such that H(a(1))  H(y) < H(k)  H(a(1)); and for n > 2, let a(n) = least k such that greatest such H(a(n1))  H(a(n2)) < H(k)  H(a(n1)). The increasing sequences H(a(n))  H(a(n1)) and a(n)/a(n1) converge. For what choices of (x,y) is the sequence a(n) linearly recurrent?
For A227965, (x,y) = (1,2); H(a(n))  H(a(n1)) approaches a limit 1.540684... given by A227966, and a(n)/a(n1) approaches a limit 4.6677834... given by A227967. It is unknown whether the sequence a(n) is linearly recurrent.


LINKS

Clark Kimberling, Table of n, a(n) for n = 1..100


EXAMPLE

The first two values (a(1),a(2)) = (11,53) match the beginning of the following inequality chain (and partition of the harmonic numbers): 1/1 + 1/2 < 1/3 + ... + 1/11 < 1/12 + ... + 1/53 < ...


MATHEMATICA

z = 300; h[n_] := h[n] = HarmonicNumber[N[n, 500]]; x = 1; y = 2;
a[1] = Ceiling[w /. FindRoot[h[w] == 2 h[y]  h[x  1], {w, 1}, WorkingPrecision > 400]]; a[2] = Ceiling[w /. FindRoot[h[w] == 2 h[a[1]]  h[y], {w, a[1]}, WorkingPrecision > 400]]; Do[s = 0; a[t] = Ceiling[w /. FindRoot[h[w] == 2 h[a[t  1]]  h[a[t  2]], {w, a[t  1]}, WorkingPrecision > 400]], {t, 3, z}];
m = Map[a, Range[z]] (* A227965 *)
t = N[Table[h[a[t]]  h[a[t  1]], {t, 2, z, 25}], 60]
Last[RealDigits[t, 10]] (* A227966 *)
t = N[Table[a[t]/a[t  1], {t, 2, z, 50}], 60]
Last[RealDigits[t, 10]] (* A227967 *)
(* A227965, Peter J. C. Moses, Jul 12 2013*)


CROSSREFS

Cf. A224820, A224868, A227728, A227966, A227967.
Sequence in context: A207361 A093864 A139967 * A196468 A252833 A227255
Adjacent sequences: A227962 A227963 A227964 * A227966 A227967 A227968


KEYWORD

nonn,frac,easy


AUTHOR

Clark Kimberling, Aug 01 2013


STATUS

approved



