A308261 For any integer n, let d(n) be the smallest k > 0 such that at least one of n-k or n+k is a prime number; we build an undirected graph G on top of the prime numbers as follows: two consecutive prime numbers p and q are connected iff at least one of d(p) or d(q) equals q-p; a(n) is the number of terms in the n-th connected component of G (ordered by least element).

4, 2, 3, 2, 7, 3, 3, 3, 3, 2, 2, 8, 2, 7, 2, 5, 4, 4, 2, 4, 5, 3, 2, 2, 3, 4, 3, 3, 2, 2, 5, 8, 7, 4, 2, 5, 3, 2, 2, 2, 2, 3, 4, 4, 3, 5, 4, 2, 2, 2, 3, 2, 3, 6, 3, 2, 2, 4, 6, 2, 3, 2, 4, 3, 4, 2, 5, 4, 3, 7, 4, 2, 2, 2, 3, 4, 4, 4, 2, 5, 4, 2, 2, 5, 3, 3, 2

Offset: 1

Author

Rémy Sigrist, Jun 02 2019

Keywords

• nonn

Comments

Each connected component of G has at least two elements.

Is the sequence bounded?

Examples

			The first terms, alongside the corresponding components, are:
  n  a(n)   n-th component
  -- ----   --------------
   1    4   {2, 3, 5, 7}
   2    2   {11, 13}
   3    3   {17, 19, 23}
   4    2   {29, 31}
   5    7   {37, 41, 43, 47, 53, 59, 61}
   6    3   {67, 71, 73}
   7    3   {79, 83, 89}
   8    3   {97, 101, 103}
   9    3   {107, 109, 113}
  10    2   {127, 131}
		

Crossrefs

Cf. A000040, A051700.

Programs

PARI

d(p) = for (k=1, oo, if (p-k>0 && isprime(p-k), return (k), isprime(p+k), return (k)))
v=1; p=2; forprime (q=p+1, oo, if (d(p)==q-p || d(q)==q-p, v++, print1 (v", "); if (n++==87, break); v = 1); p=q)