A206815 Position of n+pi(n) in the joint ranking of {j+pi(j)} and {k+(k+1)/log(k+1)}.

1, 2, 5, 7, 9, 11, 14, 15, 17, 19, 22, 23, 26, 28, 30, 31, 34, 36, 39, 40, 42, 44, 47, 49, 50, 52, 54, 56, 58, 60, 63, 65, 67, 68, 70, 72, 75, 77, 78, 80, 83, 85, 87, 89, 91, 93, 96, 98, 99, 101, 103, 105, 108, 109, 111, 113, 115, 117, 119, 121, 124, 126, 128

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, A206818 (complement of A206815).

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 *)