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

Author

Carl R. White, Jul 04 2012

Keywords

• nonn
• easy

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.

Examples

			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.
		

Links

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

Crossrefs

Cf. A214124, A214125

Programs

Mathematica

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