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

%I A377612 #8 Nov 14 2024 12:10:59
%S A377612 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,
%T A377612 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,
%U A377612 1,1,1,1,1,1,17,1,1,1,1,1,1,1,1,11,1,3
%N A377612 a(n) is the number of iterations of x -> 2*x + 1 until (# composites reached) = (# primes reached), starting with prime(n).
%C A377612 For a guide to related sequences, see A377609.
%e A377612 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.)
%t A377612 chain[{start_, u_, v_}] := NestWhile[Append[#, u*Last[#] + v] &, {start}, !
%t A377612      Count[#, _?PrimeQ] == Count[#, _?(! PrimeQ[#] &)] &];
%t A377612 chain[{Prime[1], 2, 1}]
%t A377612 Map[Length[chain[{Prime[#], 2, 1}]] &, Range[100]] - 1
%t A377612 (* _Peter J. C. Moses_, Oct 31 2024 *)
%Y A377612 Cf. A377609.
%K A377612 nonn
%O A377612 1,1
%A A377612 _Clark Kimberling_, Nov 05 2024