A377613 a(n) is the number of iterations of x -> 2*x + 3 until (# composites reached) = (# primes reached), starting with prime(n).

19, 1, 15, 15, 1, 13, 13, 15, 1, 3, 1, 1, 1, 7, 27, 3, 1, 1, 25, 1, 3, 1, 1, 5, 23, 1, 1, 1, 1, 7, 3, 1, 23, 3, 1, 1, 9, 1, 17, 5, 1, 1, 1, 3, 19, 7, 1, 3, 3, 3, 1, 1, 1, 1, 1, 1, 3, 1, 21, 1, 3, 1, 19, 1, 1, 1, 1, 3, 1, 3, 3, 1, 1, 1, 3, 3, 1, 17, 1, 3, 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+3 = 7, then 2*7+3 = 17, etc., resulting in a chain 2, 7, 17, 37, 77, 157, 317, 637, 1277, 2557, 5117, 10237, 20477, 40957, 81917, 163837, 327677, 655357, 1310717, 2621437 having 10 primes and 10 composites. Since every initial subchain has fewer composites than primes, a(1) = 20-1 = 19. (For more terms from the mapping x -> 2*x+3, see A154117.)
		

Crossrefs

Cf. A377609, A154117.

Programs

Mathematica

chain[{start_, u_, v_}] := NestWhile[Append[#, u*Last[#] + v] &, {start}, !
     Count[#, ?PrimeQ] == Count[#, ?(! PrimeQ[#] &)] &];
chain[{Prime[1], 2, 3}]
Map[Length[chain[{Prime[#], 2, 3}]] &, Range[100]] - 1
(* Peter J. C. Moses Oct 31 2024 *)