A207384 - OEIS

A207384

A206815(n+1)-A206815(n).

1, 3, 2, 2, 2, 3, 1, 2, 2, 3, 1, 3, 2, 2, 1, 3, 2, 3, 1, 2, 2, 3, 2, 1, 2, 2, 2, 2, 2, 3, 2, 2, 1, 2, 2, 3, 2, 1, 2, 3, 2, 2, 2, 2, 2, 3, 2, 1, 2, 2, 2, 3, 1, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 1, 2, 3, 2, 2, 1, 3, 2, 3, 2, 1, 2, 2, 2, 3, 2, 1, 2, 3, 2, 2, 1, 2, 2, 3, 2, 2, 1, 2, 2, 2, 2, 3, 1, 2, 2

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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]]];