A206911 - OEIS

%I #12 Jul 12 2012 00:40:00

%S 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 69,72,74,77,80,83,86,88,91,94,97,100,102,105,108,111,113,116,119,122,

%U 125,127,130,133,136,138,141,142,143,144,145,146,147,148,149

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

%e Let S(n)=1+1/2+1/3+...+1/n and L(n)=log(n+1).  Then

%e L(1)<S(1)<L(2)<L(3)<S(2)<L(4)<L(5)<S(3)<L(6)<..., so that

%e A206911=(2,5,8,...).

%t f[n_] := Sum[1/k, {k, 1, n}];  z = 300;

%t g[n_] := N[Log[n + 1]];

%t c = Table[f[n], {n, 1, z}];

%t s = Table[g[n], {n, 1, z}];

%t j = Sort[Union[c, s]];

%t p[n_] := Position[j, f[n]]; q[n_] := Position[j, g[n]];

%t Flatten[Table[p[n], {n, 1, z}]]    (* A206911 *)

%t Flatten[Table[q[n], {n, 1, z}]]    (* A206912 *)

%Y Cf. A206912, A206815.

%K nonn

%O 1,1

%A _Clark Kimberling_, Feb 13 2012