A227967 -id:A227967 - cp's OEIS frontend

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(n-1)) - H(a(n-2)) > H(k) - H(a(n-1)), where H = harmonic number.

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

Offset: 1

Clark Kimberling, Aug 01 2013

Suppose that x and y are positive integers and that x <=y. Let a(1) = least k such that H(y) - H(x-1) < 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(n-1)) - H(a(n-2)) < H(k) - H(a(n-1)). The increasing sequences H(a(n)) - H(a(n-1)) and a(n)/a(n-1) 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(n-1)) approaches a limit 1.540684... given by A227966, and a(n)/a(n-1) approaches a limit 4.6677834... given by A227967. It is unknown whether the sequence a(n) is linearly recurrent.

			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 < ...

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

Cf. A224820, A224868, A227728, A227966, A227967.

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*)

A227966 Decimal expansion of limit of H(c(n)) - H(c(n-1)), where c = A227965 and H = harmonic number.

Original entry on oeis.org

1, 5, 4, 0, 6, 8, 4, 3, 2, 6, 4, 8, 6, 0, 7, 1, 6, 4, 5, 0, 6, 1, 5, 6, 1, 5, 0, 7, 2, 9, 3, 9, 2, 0, 5, 2, 0, 1, 8, 7, 5, 2, 5, 9, 6, 7, 2, 6, 2, 2, 8, 6, 5, 7, 3, 5, 4, 4, 0, 0

Offset: 1

Clark Kimberling, Aug 01 2013

See A227965.

			1.54068432648607164506156150729392052018...

Cf. A001008, A002805 (numerator and denominator of harmonic numbers).

Cf. A227965, A227967.

Mathematica
```
(* See A227965. *)
```

cp's OEIS Frontend

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(n-1)) - H(a(n-2)) > H(k) - H(a(n-1)), where H = harmonic number.

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