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allocated for Clark Kimberling

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

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, 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, 1, 1, 1, 1, 1, 1, 17, 1, 1, 1, 1, 1, 1, 1, 1, 11, 1, 3

1,1

For a guide to related sequences, see A377609.

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.)

chain[{start_, u_, v_}] := NestWhile[Append[#, u*Last[#] + v] &, {start}, !

Count[#, _?PrimeQ] == Count[#, _?(! PrimeQ[#] &)] &];

chain[{Prime[1], 2, 1}]

Map[Length[chain[{Prime[#], 2, 1}]] &, Range[100]] - 1

(* Peter J. C. Moses Oct 31 2024 *)

Cf. A377609.

allocated

nonn

Clark Kimberling, Nov 05 2024

approved

editing

allocated for Clark Kimberling

a(n) is the number of iterations of x -> 2*x - 5 until (# composites reached) = (# primes reached), starting with prime(n+4).

25, 1, 19, 1, 11, 15, 1, 1, 1, 1, 13, 9, 3, 1, 1, 21, 1, 1, 1, 11, 1, 7, 1, 1, 1, 1, 1, 11, 17, 1, 3, 1, 1, 1, 1, 1, 13, 1, 1, 1, 5, 1, 1, 1, 3, 1, 3, 1, 1, 1, 9, 9, 1, 1, 1, 15, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 3, 3, 1, 3, 1, 1, 1, 7, 1, 1, 3

1,1

For a guide to related sequences, see A377609.

Starting with prime(5) = 11, we have 2*11-5 = 17, then 2*17-5 = 31, etc., resulting in a chain 11, 17, 29, 53, 101, 197, 389, 773, 1541, 3077, 6149, 12293, 24581, 49157, 98309, 196613, 393221, 786437, 1572869, 3145733, 6291461, 12582917, 25165829, 50331653, 100663301, 201326597 having 13 primes and 13 composites. Since every initial subchain has fewer composites than primes, a(1) = 26-1 = 25.

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

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

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

chain[{Prime[5], 2, -5}]

Map[Length[chain[{Prime[#], 2, -5}]] &, Range[5, 100]] - 1

(* Peter J. C. Moses Oct 31 2024 *)

Cf. A377609.

allocated

nonn

Clark Kimberling, Nov 05 2024

approved

editing

allocated for Clark Kimberling

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

13, 9, 7, 21, 7, 1, 15, 1, 5, 23, 5, 13, 1, 3, 1, 1, 3, 19, 1, 1, 11, 1, 7, 9, 1, 19, 1, 17, 7, 1, 3, 1, 1, 1, 11, 1, 5, 1, 1, 11, 3, 5, 1, 1, 15, 15, 1, 1, 3, 1, 5, 5, 1, 5, 1, 1, 1, 1, 13, 1, 1, 9, 1, 5, 3, 1, 3, 1, 1, 1, 1, 23, 1, 1, 1, 1, 1, 1, 1, 9, 3

1,1

For a guide to related sequences, see A377609.

Starting with prime(3) = 5, we have 2*5-3 = 7, then 2*7-3 = 11, etc., resulting in a chain 5, 7, 11, 19, 35, 67, 131, 259, 515, 1027, 2051, 4099, 8195, 16387 having 7 primes and 7 composites. Since every initial subchain has fewer composites than primes, a(1) = 14-1 = 13. (For more terms from the mapping x -> 2x-3, see A062709.)

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

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

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

chain[{Prime[3], 2, -3}]

Map[Length[chain[{Prime[#], 2, -3}]] &, Range[3, 100]] - 1

(* Peter J. C. Moses Oct 31 2024 *)

Cf. A062709, A377609.

allocated

nonn

Clark Kimberling, Nov 05 2024

approved

editing