A377612 a(n) is the number of iterations of x -> 2*x + 1 until (# composites reached) = (# primes reached), starting with prime(n).

15, 7, 13, 1, 11, 1, 1, 1, 7, 7, 1, 1, 5, 1, 1, 11, 1, 1, 1, 1, 1, 1, 3, 23, 1, 1, 1, 1, 1, 11, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 19, 1, 3, 1, 1, 1, 1, 1, 1, 1, 7, 3, 1, 3, 1, 1, 1, 1, 1, 3, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 17, 1, 1, 1, 1, 1, 1, 1, 1, 11, 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(1) = 2, we have 2*2+1 = 5, then 2*5+1 = 11, etc., resulting in a chain 2, 5, 11, 23, 47, 95, 191, 383, 767, 1535, 3071, 6143, 12287, 24575, 49151, 983033 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 -> 2x+1, see A055010.)
		

Crossrefs

Cf. A377609.

Programs

Mathematica

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