%I #8 May 04 2013 14:36:46

%S 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,

%T 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,

%U 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

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

%C 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?

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

%C 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.

%H Carl R. White, <a href="/A214123/b214123.txt">Table of n, a(n) for n = 2..10000</a>

%e 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.

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

%Y Cf. A214124, A214125

%K nonn,easy

%O 2,4

%A _Carl R. White_, Jul 04 2012