A206911 Position of n-th partial sum of the harmonic series when all the partial sums are jointly ranked with the set {log(k+1)}; complement of A206912.

2, 5, 8, 11, 13, 16, 19, 22, 24, 27, 30, 33, 36, 38, 41, 44, 47, 49, 52, 55, 58, 61, 63, 66, 69, 72, 74, 77, 80, 83, 86, 88, 91, 94, 97, 100, 102, 105, 108, 111, 113, 116, 119, 122, 125, 127, 130, 133, 136, 138, 141, 142, 143, 144, 145, 146, 147, 148, 149

Offset

1,1

Comments

Conjecture: the difference sequence of A206911 consists of 2s and 3s, and the ratio (number of 3s)/(number of 2s) tends to a number between 3.5 and 3.6.

Similar conjectures can be stated for difference sequences based on jointly ranked sets, such as A206903, A206906, A206928, A206805, A206812, and A206815.

Links

Table of n, a(n) for n=1..59.

Example

Let S(n)=1+1/2+1/3+...+1/n and L(n)=log(n+1).  Then
L(1)<S(1)<L(2)<L(3)<S(2)<L(4)<L(5)<S(3)<L(6)<..., so that
A206911=(2,5,8,...).

Mathematica

f[n_] := Sum[1/k, {k, 1, n}];  z = 300;
g[n_] := N[Log[n + 1]];
c = Table[f[n], {n, 1, z}];
s = Table[g[n], {n, 1, z}];
j = Sort[Union[c, s]];
p[n_] := Position[j, f[n]]; q[n_] := Position[j, g[n]];
Flatten[Table[p[n], {n, 1, z}]]    (* A206911 *)
Flatten[Table[q[n], {n, 1, z}]]    (* A206912 *)

Crossrefs

Cf. A206912, A206815.

Sequence in context: A187341 A329924 A330112 * A093609 A249118 A292643

Adjacent sequences: A206908 A206909 A206910 * A206912 A206913 A206914

Keyword

nonn

Author

Clark Kimberling, Feb 13 2012

Status

approved