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A207385 A206818(n+1)-A206818(n).

%I A207385 #5 Mar 30 2012 18:58:12
%S A207385 1,2,2,2,2,1,3,2,2,1,3,1,2,2,3,1,2,2,1,3,2,2,1,2,3,2,2,2,2,2,1,2,2,3,
%T A207385 2,2,1,2,3,2,1,2,2,2,2,2,2,1,2,3,2,2,2,1,3,2,2,2,2,2,2,1,2,2,2,3,2,1,
%U A207385 2,2,3,1,2,2,1,2,3,2,2,2,1,2,3,2,1,2,2,3,2,2,1,2,2,3,2,2,2,2,1
%N A207385 A206818(n+1)-A206818(n).
%C A207385 The sequences A206815, A206818, A207384, A207835 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 A207384 and A207835 are in the set {1,2,3}.
%e A207385 The joint ranking is represented by
%e A207385 1 < 3 < 3.8 < 4.7 < 5 < 5.8 < 6 <7.1 < 8 < 8.3 < 9 < ...
%e A207385 Positions of numbers j+pi(j): 1,2,5,7,9,...
%e A207385 Positions of numbers k+(k+1)/log(k+1): 3,4,6,8,10,..
%t A207385 f[1, n_] := n + PrimePi[n];
%t A207385 f[2, n_] := n + N[(n + 1)/Log[n + 1]]; z = 500;
%t A207385 t[k_] := Table[f[k, n], {n, 1, z}];
%t A207385 t = Sort[Union[t[1], t[2]]];
%t A207385 p[k_, n_] := Position[t, f[k, n]];
%t A207385 Flatten[Table[p[1, n], {n, 1, z}]]    (* A206815 *)
%t A207385 Flatten[Table[p[2, n], {n, 1, z}]]    (* A206818 *)
%t A207385 d1[n_] := p[1, n + 1] - p[1, n]
%t A207385 Flatten[Table[d1[n], {n, 1, z - 1}]]  (* A207385 *)
%t A207385 d2[n_] := p[2, n + 1] - p[2, n]
%t A207385 Flatten[Table[d2[n], {n, 1, z - 1}]]  (* A207386 *)
%Y A207385 Cf. A000720, A206815, A206818.
%K A207385 nonn
%O A207385 1,2
%A A207385 _Clark Kimberling_, Feb 17 2012