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

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, 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, 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

Offset

1,2

Comments

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}.

Links

Table of n, a(n) for n=1..99.

Example

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,..

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}]]  (* A207385 *)
d2[n_] := p[2, n + 1] - p[2, n]
Flatten[Table[d2[n], {n, 1, z - 1}]]  (* A207386 *)

Crossrefs

Cf. A000720, A206815, A206818.

Sequence in context: A306250 A145443 A192056 * A306242 A118144 A136691

Adjacent sequences: A207382 A207383 A207384 * A207386 A207387 A207388

Keyword

nonn

Author

Clark Kimberling, Feb 17 2012

Status

approved