A377624 a(n) is the number of iterations of x -> 6*x + 5 until (# composites reached) = (# primes reached), starting with prime(n).

17, 5, 1, 3, 9, 11, 15, 1, 1, 3, 7, 5, 5, 7, 1, 1, 5, 1, 1, 7, 5, 5, 9, 1, 5, 1, 1, 9, 3, 13, 1, 1, 5, 9, 1, 11, 5, 7, 1, 1, 1, 13, 5, 7, 9, 1, 1, 1, 3, 1, 1, 9, 3, 3, 1, 3, 7, 1, 9, 1, 1, 1, 5, 5, 1, 13, 1, 3, 9, 3, 1, 1, 3, 17, 1, 1, 5, 1, 3, 9, 1, 5, 5, 1

Offset: 1

Author

Clark Kimberling, Nov 20 2024

Keywords

• nonn

Comments

For a guide to related sequences, see A377609.

Examples

			Starting with prime(1) = 2, we have 6*2+5 = 17, then 6*17+5 = 107, etc., resulting in a chain 2, 17, 107, 647, 3887, 23327, 139967, 839807, 5038847, 30233087, 181398527, 1088391167, 6530347007, 39182082047, 235092492287, 1410554953727, 8463329722367, 50779978334207 having 9 primes and 9 composites. Since every initial subchain has fewer composites than primes, a(1) = 18-1 = 17. (For more terms from the mapping x -> 6x-5, see A198796.)
		

Crossrefs

Cf. A377609, A198796.

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[1], 6, -5}]
Map[Length[chain[{Prime[#], 6, -5}]] &, Range[1, 100]] - 1
(* Peter J. C. Moses, Oct 31 2024 *)