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

1,1

Comments

Each connected component of G has at least two elements.

Is the sequence bounded?

Links

Table of n, a(n) for n=1..87.

Example

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}

Prog

(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)

Crossrefs

Cf. A000040, A051700.

Sequence in context: A238352 A291357 A079636 * A019614 A051528 A073244

Adjacent sequences: A308258 A308259 A308260 * A308262 A308263 A308264

Keyword

nonn

Author

Rémy Sigrist, Jun 02 2019

Status

approved