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

3, 4, 6, 8, 10, 12, 13, 16, 18, 20, 21, 24, 25, 27, 29, 32, 33, 35, 37, 38, 41, 43, 45, 46, 48, 51, 53, 55, 57, 59, 61, 62, 64, 66, 69, 71, 73, 74, 76, 79, 81, 82, 84, 86, 88, 90, 92, 94, 95, 97, 100, 102, 104, 106, 107, 110, 112, 114, 116, 118, 120, 122, 123

Offset: 1

Author

Clark Kimberling, Feb 17 2012

Keywords

• nonn

Comments

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}.

Examples

			The joint ranking is represented by
1 < 3 < 3.8 < 4.7 < 5 < 5.8 < 6 <7.1 < 8 < 8.3 < 9 < ...
Positions of numbers j+pi(j): 1,2,5,7,9,...
Positions of numbers k+(k+1)/log(k+1): 3,4,6,8,10,..
		

Crossrefs

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

Programs

Mathematica

f[1, n_] := n + PrimePi[n];
f[2, n_] := n + N[(n + 1)/Log[n + 1]]; z = 500;
t[k_] := Table[f[k, n], {n, 1, z}];
t = Sort[Union[t[1], t[2]]];
p[k_, n_] := Position[t, f[k, n]];
Flatten[Table[p[1, n], {n, 1, z}]]    (* A206815 *)
Flatten[Table[p[2, n], {n, 1, z}]]    (* A206818 *)
d1[n_] := p[1, n + 1] - p[1, n]
Flatten[Table[d1[n], {n, 1, z - 1}]]  (* A206827 *)
d2[n_] := p[2, n + 1] - p[2, n]
Flatten[Table[d2[n], {n, 1, z - 1}]]  (* A206828 *)