A377621 a(n) is the number of iterations of x -> 6*x - 1 until (# composites reached) = (# primes reached), starting with prime(n).

11, 7, 7, 3, 1, 1, 3, 5, 5, 5, 1, 1, 1, 3, 3, 5, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 5, 3, 5, 3, 3, 1, 3, 1, 1, 1, 3, 5, 1, 1, 1, 1, 1, 1, 9, 3, 1, 1, 5, 7, 1, 9, 1, 1, 1, 1, 9, 1, 1, 1, 5, 1, 1, 1, 5, 3, 1, 1, 3, 1, 1, 5, 1, 7, 3, 9, 7, 3, 1, 1, 1, 1, 1, 1, 7, 3

Offset

1,1

Comments

For a guide to related sequences, see A377609.

Links

Table of n, a(n) for n=1..86.

Example

Starting with prime(1) = 2, we have 6*2-1 = 11, then 6*11-1 = 65, etc., resulting in a chain 2, 11, 65, 389, 2333, 13997, 83981, 503885, 3023309, 18139853, 108839117, 653034701 having 6 primes and 6 composites. Since every initial subchain has fewer composites than primes, a(1) = 12-1 = 11. (For more terms from the mapping x -> 6x-1, see A199412.)

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

Crossrefs

Cf. A377609, A199412.

Sequence in context: A240598 A144262 A110093 * A282345 A265765 A187563

Adjacent sequences: A377618 A377619 A377620 * A377622 A377623 A377624

Keyword

nonn

Author

Clark Kimberling, Nov 20 2024

Status

approved