_Clark Kimberling (ck6(AT)evansville.edu), _, Feb 17 2012

proposed

approved

editing

proposed

The sequences A206815, A206816, 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}.

Flatten[Table[p[2, n], {n, 1, z}]] (* A206816 A206818 *)

Cf. A000720, A206815, A206816A206818.

allocated for Clark Kimberling

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

1,2

The sequences A206815, A206816, 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}.

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,..

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}]] (* A206816 *)

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 *)

Cf. A000720, A206815, A206816.

allocated

nonn

Clark Kimberling (ck6(AT)evansville.edu), Feb 17 2012

approved

editing

allocating

allocated

allocated for Clark Kimberling

allocating

approved