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Position of n+(n+1)/log(n+1) in the joint ranking of {j+pi(j)} and {k+(k+1)/log(k+1)}.
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%I #6 Mar 30 2012 18:58:12

%S 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 48,51,53,55,57,59,61,62,64,66,69,71,73,74,76,79,81,82,84,86,88,90,92,

%U 94,95,97,100,102,104,106,107,110,112,114,116,118,120,122,123

%N Position of n+(n+1)/log(n+1) 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, A206815 (complement of A206818).

%K nonn

%O 1,1

%A _Clark Kimberling_, Feb 17 2012