A377618 - OEIS

%I #8 Nov 21 2024 11:36:29

%S 5,17,3,1,15,1,3,1,1,1,1,1,3,1,1,5,1,1,1,5,1,1,3,1,1,1,1,1,1,1,1,5,5,

%T 1,1,1,1,1,1,3,1,1,1,1,3,1,1,1,3,1,1,1,1,1,1,5,1,1,1,3,1,13,1,1,1,1,1,

%U 1,1,1,1,1,1,1,1,3,1,1,1,1,1,1,3,1,1

%N a(n) is the number of iterations of x -> 4*x - 1 until (# composites reached) = (# primes reached), starting with prime(n).

%C For a guide to related sequences, see A377609.

%e Starting with prime(1) = 2, we have 4*2-1 = 7, then 4*7-1 = 27, etc.,

%e resulting in a chain 2, 7, 27, 107, 427, 1707 having 3 primes and 3 composites. Since every initial subchain has fewer composites than primes, a(1) = 6-1 = 5. (For more terms from the mapping x -> 4x-1, see A136412.)

%t chain[{start_, u_, v_}] := If[CoprimeQ[u, v] && start*u + v != start,

%t    NestWhile[Append[#, u*Last[#] + v] &, {start}, !

%t       Count[#, _?PrimeQ] == Count[#, _?(! PrimeQ[#] &)] &], {}];

%t chain[{Prime[1], 4, -1}]

%t Map[Length[chain[{Prime[#], 4, -1}]] &, Range[1, 100]] - 1

%t (* _Peter J. C. Moses_, Oct 31 2024 *)

%Y Cf. A377609, A136412.

%K nonn

%O 1,1

%A _Clark Kimberling_, Nov 17 2024