A291654 Number of distinct values in the prime tree starting with n.

35, 35, 34, 22, 12, 11, 5, 7, 10, 5, 10, 10, 6, 6, 7, 7, 4, 3, 5, 13, 14, 6, 1, 5, 5, 3, 4, 3, 2, 2, 4, 3, 2, 2, 3, 3, 1, 4, 6, 3, 3, 3, 1, 3, 7, 6, 2, 2, 2, 6, 6, 1, 6, 9, 5, 5, 5, 2, 4, 5, 2, 2, 2, 7, 8, 7, 6, 2, 3, 4, 5, 3, 1, 1, 4, 5, 4, 4

Offset: 1

Author

Ralf Steiner and Sean A. Irvine, Aug 28 2017

Keywords

• nonn

Comments

Starting from n construct a tree that includes nodes for each prime in n^2 + n + 1, n^2 + n - 1, n^2 - n + 1, n^2 - n - 1, and recurse on each node until no further primes can be included.

Examples

			a(5) = 12 since the tree for 5 looks like this where, for example, the symbol -[+-]->  stands for p^2+p-1 and the symbol -| stands for a leaf:
5-[--]->19-[+-]->379-[--]->143261-|
-[-+]->143263-[-+]->20524143907-|
-[+-]->29-[--]->811-|
-[++]->31-[--]->929-|
-[+-]->991-[-+]->981091-|
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Ralf Steiner, Prime-trees, NMBRTHRY posting, 13 Aug 2017.

Programs

Maple

f:= proc(n) local  R, agenda;
  agenda:= {n}; R:= {n};
  while nops(agenda) > 0 do
    agenda:= select(isprime, map(t -> (t^2+t+1,t^2+t-1,t^2-t+1,t^2-t-1), agenda) minus R) ;
    R:= R union agenda;
  od;
  nops(R);
end proc:
map(f, [$1..100]); # Robert Israel, Aug 29 2017

Mathematica

f[n_] := Module[{R = {n}, agenda = {n}}, While[Length[agenda] > 0, agenda = Select[Flatten[Map[Function[t, {t^2 + t + 1, t^2 + t - 1, t^2 - t + 1, t^2 - t - 1}], agenda]] ~Complement~ R, PrimeQ]; R = Union[R, agenda]]; Length[R]];
Array[f, 100] (* Jean-François Alcover, Jul 30 2020, after Robert Israel *)