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A377618 a(n) is the number of iterations of x -> 4*x - 1 until (# composites reached) = (# primes reached), starting with prime(n).

%I A377618 #8 Nov 21 2024 11:36:29
%S A377618 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 A377618 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 A377618 1,1,1,1,1,1,1,1,3,1,1,1,1,1,1,3,1,1
%N A377618 a(n) is the number of iterations of x -> 4*x - 1 until (# composites reached) = (# primes reached), starting with prime(n).
%C A377618 For a guide to related sequences, see A377609.
%e A377618 Starting with prime(1) = 2, we have 4*2-1 = 7, then 4*7-1 = 27, etc.,
%e A377618 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 A377618 chain[{start_, u_, v_}] := If[CoprimeQ[u, v] && start*u + v != start,
%t A377618    NestWhile[Append[#, u*Last[#] + v] &, {start}, !
%t A377618       Count[#, _?PrimeQ] == Count[#, _?(! PrimeQ[#] &)] &], {}];
%t A377618 chain[{Prime[1], 4, -1}]
%t A377618 Map[Length[chain[{Prime[#], 4, -1}]] &, Range[1, 100]] - 1
%t A377618 (* _Peter J. C. Moses_, Oct 31 2024 *)
%Y A377618 Cf. A377609, A136412.
%K A377618 nonn
%O A377618 1,1
%A A377618 _Clark Kimberling_, Nov 17 2024