A364787 a(n) is the stabilization index of the prime ladder [P(n,k) : k >= 0].

0, 1, 3, 2, 7, 6, 17, 17, 19, 18, 13, 13, 11, 11, 47, 46, 39, 39, 59, 59, 68, 68, 71, 71, 61, 61, 60, 59, 56, 55, 49, 49, 47, 47, 334, 333, 508, 508, 488, 488, 466, 466, 423, 423, 512, 512, 488, 488, 468, 468, 450, 450, 696, 696, 652, 652, 639, 638, 613, 613

Offset: 0

Author

Eduard Roure Perdices, Aug 07 2023

Keywords

• nonn

Comments

Given n >= 0, we consider the following increasing sequence of prime numbers: P(n,0) = 2, and for k > 0, P(n,k) is the largest prime number smaller than or equal to P(n,k-1)+n. Since the sequence of all prime numbers has arbitrarily long gaps, there exists an index m >= 0 such that P(n,m) = P(n,m+1). We define a(n) as the smallest of such indices.

Note that a(n) displays big jumps at values of n corresponding to maximal prime gaps (A005250).

In general, for k >= 0, a(2k+1) = a(2k), but there are exceptions: for n = 0, 2, 4, 8, 14, 26, 28, 34, 56, 94, 154, and 484, |a(n+1) - a(n)| = 1. We don't know if there are more of these blips.

Examples

			a(4) = 7 because P(4,0) = 2, P(4,1) = 5, P(4,2) = 7, P(4,3) = 11, P(4,4) = 13, P(4,5) = 17, P(4,6) = 19, and for k >= 7, P(4,k) = 23.
		

Links

• Eduard Roure Perdices, Table of n, a(n) for n = 0..807

Crossrefs

Cf. A000040, A001223, A002386, A005250.

Programs

Mathematica

SequenceA[n_] := Module[{pn0 = 2, pnk, an = 0},
   While[True, pnk = NextPrime[pn0 + n + 1, -1];
    If[pn0 == pnk, Break[], pn0 = pnk; an++]];
   Return[an];];