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A206818 Position of n+(n+1)/log(n+1) in the joint ranking of {j+pi(j)} and {k+(k+1)/log(k+1)}.

%I A206818 #6 Mar 30 2012 18:58:12
%S A206818 3,4,6,8,10,12,13,16,18,20,21,24,25,27,29,32,33,35,37,38,41,43,45,46,
%T A206818 48,51,53,55,57,59,61,62,64,66,69,71,73,74,76,79,81,82,84,86,88,90,92,
%U A206818 94,95,97,100,102,104,106,107,110,112,114,116,118,120,122,123
%N A206818 Position of n+(n+1)/log(n+1) in the joint ranking of {j+pi(j)} and {k+(k+1)/log(k+1)}.
%C A206818 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 A206818 The joint ranking is represented by
%e A206818 1 < 3 < 3.8 < 4.7 < 5 < 5.8 < 6 <7.1 < 8 < 8.3 < 9 < ...
%e A206818 Positions of numbers j+pi(j): 1,2,5,7,9,...
%e A206818 Positions of numbers k+(k+1)/log(k+1): 3,4,6,8,10,..
%t A206818 f[1, n_] := n + PrimePi[n];
%t A206818 f[2, n_] := n + N[(n + 1)/Log[n + 1]]; z = 500;
%t A206818 t[k_] := Table[f[k, n], {n, 1, z}];
%t A206818 t = Sort[Union[t[1], t[2]]];
%t A206818 p[k_, n_] := Position[t, f[k, n]];
%t A206818 Flatten[Table[p[1, n], {n, 1, z}]]    (* A206815 *)
%t A206818 Flatten[Table[p[2, n], {n, 1, z}]]    (* A206818 *)
%t A206818 d1[n_] := p[1, n + 1] - p[1, n]
%t A206818 Flatten[Table[d1[n], {n, 1, z - 1}]]  (* A206827 *)
%t A206818 d2[n_] := p[2, n + 1] - p[2, n]
%t A206818 Flatten[Table[d2[n], {n, 1, z - 1}]]  (* A206828 *)
%Y A206818 Cf. A000720, A206827, A206815 (complement of A206818).
%K A206818 nonn
%O A206818 1,1
%A A206818 _Clark Kimberling_, Feb 17 2012