A377623 a(n) is the number of iterations of x -> 6*x + 1 until (# composites reached) = (# primes reached), starting with prime(n).

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

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 = 13, then 6*13+1 = 79, etc., resulting in a chain 2, 13, 79, 475, 2851, 17107, 102643, 615859, 3695155, 22170931, 133025587, 798153523, 4788921139, 28733526835, 172401161011, 1034406966067 having 8 primes and 8 composites. Since every initial subchain has fewer composites than primes, a(1) = 16-1 = 15. (For more terms from the mapping x -> 6x-5, see A198849.)

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 *)

Crossrefs

Cf. A377609, A198849.

Sequence in context: A296009 A104436 A070601 * A332843 A196187 A004480

Adjacent sequences: A377620 A377621 A377622 * A377624 A377625 A377626

Keyword

nonn

Author

Clark Kimberling, Nov 20 2024

Status

approved