A226896 Position of log n in the joint ranking of harmonic numbers H(k) and {log k}, for k >= 1; complement of A226894.

1, 2, 4, 5, 7, 8, 10, 11, 13, 15, 16, 18, 19, 21, 22, 24, 26, 27, 29, 30, 32, 33, 35, 36, 38, 40, 41, 43, 44, 46, 47, 49, 51, 52, 54, 55, 57, 58, 60, 61, 63, 65, 66, 68, 69, 71, 72, 74, 76, 77, 79, 80, 82, 83, 85, 86, 88, 90, 91, 93, 94, 96, 97, 99, 100, 102

Offset

1,2

Comments

If, in the definition, log k is replaced by g + log k, where g is the Euler-Mascheroni constant, then the position of log n is 2n-1, which leads to a conjecture: limit[1/(H(n) - g - log n) - 2n] = 1/3.

Links

Clark Kimberling, Table of n, a(n) for n = 1..1000

Example

log 1 < log 2 < H(1) < log 3 < log 4 < H(2) < ...

Mathematica

z = 300; h[n_] := N[HarmonicNumber[n], 100]; t1 = Table[h[n], {n, 1, z}]; t2 = Table[N[Log[n], 100], {n, 1, 3 z}]; t3 = Union[t1, t2]; p[n_] := Position[t3, h[n]]; Flatten[Table[p[n], {n, 1, 3 z}]] (* A226894 *)
Differences[%]  (* A226895 *)
Complement[Range[z], %%]  (* A226896 *)

Crossrefs

Cf. A001008(n)/A002805(n) (H(n)), A226894.

Sequence in context: A161750 A249400 A347782 * A175302 A288521 A067940

Adjacent sequences: A226893 A226894 A226895 * A226897 A226898 A226899

Keyword

nonn,easy

Author

Clark Kimberling, Jun 21 2013

Status

approved