A377611 a(n) is the number of iterations of x -> 2*x - 5 until (# composites reached) = (# primes reached), starting with prime(n+4).

25, 1, 19, 1, 11, 15, 1, 1, 1, 1, 13, 9, 3, 1, 1, 21, 1, 1, 1, 11, 1, 7, 1, 1, 1, 1, 1, 11, 17, 1, 3, 1, 1, 1, 1, 1, 13, 1, 1, 1, 5, 1, 1, 1, 3, 1, 3, 1, 1, 1, 9, 9, 1, 1, 1, 15, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 3, 3, 1, 3, 1, 1, 1, 7, 1, 1, 3

Offset: 1

Author

Clark Kimberling, Nov 05 2024

Keywords

• nonn

Comments

For a guide to related sequences, see A377609.

Examples

			Starting with prime(5) = 11, we have 2*11-5 = 17, then 2*17-5 = 31, etc., resulting in a chain 11, 17, 29, 53, 101, 197, 389, 773, 1541, 3077, 6149, 12293, 24581, 49157, 98309, 196613, 393221, 786437, 1572869, 3145733, 6291461, 12582917, 25165829, 50331653, 100663301, 201326597 having 13 primes and 13 composites. Since every initial subchain has fewer composites than primes, a(1) = 26-1 = 25.
		

Crossrefs

Cf. A377609.

Programs

Mathematica

chain[{start_, u_, v_}] := If[CoprimeQ[u, v] && start*u + v != start,
   NestWhile[Append[#, u*Last[#] + v] &, {start}, !
      Count[#, ?PrimeQ] == Count[#, ?(! PrimeQ[#] &)] &], {}];
chain[{Prime[5], 2, -5}]
Map[Length[chain[{Prime[#], 2, -5}]] &, Range[5, 100]] - 1
(* Peter J. C. Moses, Oct 31 2024 *)