cp's OEIS Frontend

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

%I A214123 #10 Jul 22 2025 23:09:31
%S A214123 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 A214123 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 A214123 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 A214123 Smallest positive k such that n+k(n-1) is prime.
%C A214123 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 A214123 Where the minimum is allowed to be zero, substitute a(n) = 0 for prime n.
%C A214123 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 A214123 Carl R. White, <a href="/A214123/b214123.txt">Table of n, a(n) for n = 2..10000</a>
%e A214123 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 A214123 spk[n_]:=Module[{k=1},While[!PrimeQ[n+k(n-1)],k++];k]; Array[spk,150,2] (* _Harvey P. Dale_, May 04 2013 *)
%Y A214123 Cf. A214124, A214125
%K A214123 nonn,easy
%O A214123 2,4
%A A214123 _Carl R. White_, Jul 04 2012