cp's OEIS Frontend

A377609 a(n) is the number of iterations of x -> 2*x - 1 until (# composites reached) = (# primes reached), starting with prime(n).

%I A377609 #10 Nov 14 2024 12:11:10
%S A377609 7,5,1,3,1,1,1,9,1,1,5,3,1,1,1,1,1,1,1,1,1,19,1,1,5,1,1,1,1,1,1,1,1,3,
%T A377609 1,1,7,1,1,1,1,1,1,1,1,3,3,1,1,3,1,1,1,1,1,1,1,7,1,1,1,1,3,1,1,1,13,7,
%U A377609 1,1,1,1,3,1,3,1,1,1,1,1,1,1,1,1,5,1
%N A377609 a(n) is the number of iterations of x -> 2*x - 1 until (# composites reached) = (# primes reached), starting with prime(n).
%C A377609 This sequence represents a family of sequences (s(n)) defined as follows: suppose that u and v are fixed coprime integers, with u >= 2. Let s(n) be the number of iterations of x -> u*x + v until (# composites reached) = (# primes reached), starting with prime(n).
%C A377609 In the following guide to related sequences LIC abbreviates "length of initial chain":
%C A377609 sequence  1st term   generator     LIC
%C A377609 A377609       2      x -> 2x-1      8
%C A377609 A377610       5      x -> 2x-3     14
%C A377609 A377611      11      x -> 2x-5     26
%C A377609 A377612       2      x -> 2x+1     16
%C A377609 A377613       2      x -> 2x+3     20
%C A377609 A377614       2      x -> 2x+5      2
%C A377609 A377615       2      x -> 2x+7     24
%C A377609 A377616       2      x -> 3x+2      2
%C A377609 A377617       2      x -> 3x+4      2
%C A377609 A377618       2      x -> 4x-1      6
%C A377609 A377619       2      x -> 5x+2      2
%C A377609 A377620       2      x -> 5x+4      2
%C A377609 A377621       2      x -> 6x-1      2
%C A377609 A377622       2      x -> 6x-5     12
%C A377609 A377623       2      x -> 6x+1     16
%C A377609 A377624       2      x -> 6x+5     18
%e A377609 Starting with prime(1) = 2, we have 2*2-1 = 3, then 2*3-1 = 5, etc., resulting in a chain 2 -> 3 -> 5 -> 9 -> 17 -> 33 -> 65 -> 129. Writing p for primes and c for nonprimes, the chain gives p, p, p, c, p, c, c, c, so that a(1) = 7, since it takes 7 arrows for the number of c's to catch up to the number of p's. (For more terms from the mapping x -> 2x-1, see A000051.)
%t A377609 chain[{start_, u_, v_}] := NestWhile[Append[#, u*Last[#] + v] &, {start}, !
%t A377609      Count[#, _?PrimeQ] == Count[#, _?(! PrimeQ[#] &)] &];
%t A377609 chain[{Prime[1], 2, -1}]
%t A377609 Map[Length[chain[{Prime[#], 2, -1}]] &, Range[100]] - 1
%t A377609 (* _Peter J. C. Moses_, Oct 31 2024 *)
%Y A377609 Cf. A000040, A000051, A377610-A377624.
%K A377609 nonn
%O A377609 1,1
%A A377609 _Clark Kimberling_, Nov 05 2024