A206815 - OEIS

%I #6 Mar 30 2012 18:58:12

%S 1,2,5,7,9,11,14,15,17,19,22,23,26,28,30,31,34,36,39,40,42,44,47,49,

%T 50,52,54,56,58,60,63,65,67,68,70,72,75,77,78,80,83,85,87,89,91,93,96,

%U 98,99,101,103,105,108,109,111,113,115,117,119,121,124,126,128

%N Position of n+pi(n) in the joint ranking of {j+pi(j)} and {k+(k+1)/log(k+1)}.

%C The sequences A206815, A206818, A206827, A206828 illustrate the closeness of {j+pi(j)} to {k+(k+1)/log(k+1)}, as suggested by the prime number theorem and the conjecture that all the terms of A206827 and A206828 are in the set {1,2,3}.

%e The joint ranking is represented by

%e 1 < 3 < 3.8 < 4.7 < 5 < 5.8 < 6 <7.1 < 8 < 8.3 < 9 < ...

%e Positions of numbers j+pi(j): 1,2,5,7,9,...

%e Positions of numbers k+(k+1)/log(k+1): 3,4,6,8,10,..

%t f[1, n_] := n + PrimePi[n];

%t f[2, n_] := n + N[(n + 1)/Log[n + 1]]; z = 500;

%t t[k_] := Table[f[k, n], {n, 1, z}];

%t t = Sort[Union[t[1], t[2]]];

%t p[k_, n_] := Position[t, f[k, n]];

%t Flatten[Table[p[1, n], {n, 1, z}]]    (* A206815 *)

%t Flatten[Table[p[2, n], {n, 1, z}]]    (* A206818 *)

%t d1[n_] := p[1, n + 1] - p[1, n]

%t Flatten[Table[d1[n], {n, 1, z - 1}]]  (* A206827 *)

%t d2[n_] := p[2, n + 1] - p[2, n]

%t Flatten[Table[d2[n], {n, 1, z - 1}]]  (* A206828 *)

%Y Cf. A000720, A206827, A206818 (complement of A206815).

%K nonn

%O 1,2

%A _Clark Kimberling_, Feb 17 2012