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A364787
a(n) is the stabilization index of the prime ladder [P(n,k) : k >= 0].
1
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
0,3
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.
LINKS
Eduard Roure Perdices, Table of n, a(n) for n = 0..807
EXAMPLE
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.
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]; ];
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved