cp's OEIS Frontend

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.

%I A206911 #12 Jul 12 2012 00:40:00
%S A206911 2,5,8,11,13,16,19,22,24,27,30,33,36,38,41,44,47,49,52,55,58,61,63,66,
%T A206911 69,72,74,77,80,83,86,88,91,94,97,100,102,105,108,111,113,116,119,122,
%U A206911 125,127,130,133,136,138,141,142,143,144,145,146,147,148,149
%N 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.
%C A206911 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.
%C A206911 Similar conjectures can be stated for difference sequences based on jointly ranked sets, such as A206903, A206906, A206928, A206805, A206812, and A206815.
%e A206911 Let S(n)=1+1/2+1/3+...+1/n and L(n)=log(n+1).  Then
%e A206911 L(1)<S(1)<L(2)<L(3)<S(2)<L(4)<L(5)<S(3)<L(6)<..., so that
%e A206911 A206911=(2,5,8,...).
%t A206911 f[n_] := Sum[1/k, {k, 1, n}];  z = 300;
%t A206911 g[n_] := N[Log[n + 1]];
%t A206911 c = Table[f[n], {n, 1, z}];
%t A206911 s = Table[g[n], {n, 1, z}];
%t A206911 j = Sort[Union[c, s]];
%t A206911 p[n_] := Position[j, f[n]]; q[n_] := Position[j, g[n]];
%t A206911 Flatten[Table[p[n], {n, 1, z}]]    (* A206911 *)
%t A206911 Flatten[Table[q[n], {n, 1, z}]]    (* A206912 *)
%Y A206911 Cf. A206912, A206815.
%K A206911 nonn
%O A206911 1,1
%A A206911 _Clark Kimberling_, Feb 13 2012