A377618 a(n) is the number of iterations of x -> 4*x - 1 until (# composites reached) = (# primes reached), starting with prime(n).

5, 17, 3, 1, 15, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 5, 1, 1, 1, 5, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 5, 5, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 3, 1, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 3, 1, 1

Offset: 1

Author

Clark Kimberling, Nov 17 2024

Keywords

• nonn

Comments

For a guide to related sequences, see A377609.

Examples

			Starting with prime(1) = 2, we have 4*2-1 = 7, then 4*7-1 = 27, etc.,
resulting in a chain 2, 7, 27, 107, 427, 1707 having 3 primes and 3 composites. Since every initial subchain has fewer composites than primes, a(1) = 6-1 = 5. (For more terms from the mapping x -> 4x-1, see A136412.)
		

Crossrefs

Cf. A377609, A136412.

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], 4, -1}]
Map[Length[chain[{Prime[#], 4, -1}]] &, Range[1, 100]] - 1
(* Peter J. C. Moses, Oct 31 2024 *)