A377615 a(n) is the number of iterations of x -> 2*x + 7 until (# composites reached) = (# primes reached), starting with prime(n).

23, 7, 9, 1, 21, 1, 7, 1, 21, 1, 1, 1, 3, 1, 3, 19, 1, 1, 1, 5, 1, 1, 7, 1, 1, 1, 1, 1, 1, 17, 1, 9, 17, 1, 1, 1, 1, 1, 1, 5, 1, 1, 3, 1, 15, 1, 1, 1, 9, 1, 1, 1, 1, 3, 17, 1, 1, 1, 1, 15, 1, 11, 1, 1, 1, 5, 1, 1, 11, 1, 1, 1, 1, 1, 1, 23, 1, 1, 11, 1, 1, 1

Offset: 1

Author

Clark Kimberling, Nov 13 2024

Keywords

• nonn

Comments

For a guide to related sequences, see A377609.

Examples

			Starting with prime(1) = 2, we have 2*2+7 = 11, then 2*11+7 = 29, etc., resulting in a chain 2, 11, 29, 65, 137, 281, 569, 1145, 2297, 4601, 9209, 18425, 36857, 73721, 147449, 294905, 589817, 1179641, 2359289, 4718585, 9437177, 18874361, 37748729, 75497465 having 24 primes and 24 composites. Since every initial subchain has fewer composites than primes, a(1) =   24-1 = 23. (For more terms from the mapping x -> 2x+7, see A154251.)
		

Crossrefs

Cf. A377609, A154251.

Programs

Mathematica

chain[{start_, u_, v_}] := NestWhile[Append[#, u*Last[#] + v] &, {start}, !
     Count[#, ?PrimeQ] == Count[#, ?(! PrimeQ[#] &)] &];
chain[{Prime[1], 2, 7}]
Map[Length[chain[{Prime[#], 2, 7}]] &, Range[100]] - 1
(* Peter J. C. Moses Oct 31 2024 *)