A214123 Smallest positive k such that n+k(n-1) is prime

1, 1, 1, 2, 1, 1, 3, 1, 1, 2, 1, 2, 3, 1, 1, 5, 5, 1, 9, 1, 1, 2, 1, 2, 3, 1, 3, 3, 1, 1, 9, 2, 1, 2, 1, 1, 3, 4, 1, 5, 1, 2, 3, 1, 3, 2, 5, 1, 3, 1, 1, 2, 1, 1, 5, 1, 3, 3, 11, 2, 5, 4, 1, 2, 1, 2, 3, 1, 1, 2, 7, 5, 3, 1, 1, 2, 5, 1, 3, 2, 1, 8, 1, 3, 11, 1, 3, 3, 1, 1, 5, 2, 3, 2, 1, 1, 3, 1, 1, 3, 5, 2, 5, 2, 1, 6, 5, 3, 9, 2, 1, 2, 1, 1, 3, 1, 7, 5, 1, 1, 5, 2, 5, 2, 1, 2, 3, 1, 7, 3, 1, 2, 11, 1, 1, 2, 5, 1, 3, 1, 1, 3, 5, 2, 9, 1, 5, 3

Offset

2,4

Comments

Given n fenceposts, what is the minimum (but greater than zero) number of new posts which can be inserted between each consecutive pair of original posts to obtain a prime number of total posts?

Where the minimum is allowed to be zero, substitute a(n) = 0 for prime n.

a(n) is 1 when 2n-1 is prime, which is equivalent to a((p+1)/2)=1 for prime p > 2, therefore there are an infinite number of pairs of consecutive 1s in the sequence if the twin prime conjecture is true.

Links

Carl R. White, Table of n, a(n) for n = 2..10000

Example

For n = 5, we have fenceposts like so: ||||| . To insert 1 post between each pair of original posts would leave us with 9 posts: |;|;|;|;|, which is not prime. Inserting two: |;;|;;|;;|;;| gives 13 posts. This is prime so a(5) = 2.

Mathematica

spk[n_]:=Module[{k=1}, While[!PrimeQ[n+k(n-1)], k++]; k]; Array[spk, 150, 2] (* Harvey P. Dale, May 04 2013 *)

Crossrefs

Cf. A214124, A214125

Sequence in context: A137457 A275699 A305633 * A369188 A007862 A285851

Adjacent sequences: A214120 A214121 A214122 * A214124 A214125 A214126

Keyword

nonn,easy

Author

Carl R. White, Jul 04 2012

Status

approved