A216189 Numbers n whose odd Collatz steps (except for 1) are all primes.

3, 5, 6, 7, 9, 10, 11, 12, 13, 14, 17, 19, 20, 21, 22, 24, 25, 26, 28, 29, 34, 35, 37, 38, 39, 40, 44, 45, 48, 49, 52, 53, 56, 58, 59, 67, 68, 69, 74, 76, 77, 80, 85, 87, 88, 89, 96, 99, 101, 104, 106, 112, 116, 117, 118, 119, 131, 134, 136, 141, 148, 149

Offset: 1

Author

Alessandro Polcini, Oct 10 2018

Keywords

• nonn

Comments

Powers of 2 are not included as they don't have odd Collatz steps. The number n itself (if odd) is not counted as an odd step. [Corrected by Jianing Song, Dec 09 2018]

Examples

			7 is in this sequence because 7*3+1 = 22; 22/2 = [11]; 11*3+1 = 34; 34/2 = [17]; 17*3+1 = 52; 52/2 = 26; 26/2 = [13]; 13*3+1 = 40; 40/2 = 20; 20/2 = 10; 10/2 = [5]; 5*3+1 = 16; 16/2 = 8; 8/2 = 4; 4/2 = 2; 2/2 = 1. 11, 17, 13 and 5 are all primes.
15 is not in this sequence because 15*3+1 = 46; 46/2 = [23]; 23*3+1 = 70; 70/2 = 35, which isn't a prime number.
		

Links

• Index entries for sequences related to 3x+1 (or Collatz) problem

Crossrefs

Cf. A006577, A177000.

Programs

Mathematica

Select[Range[3, 150], And[AllTrue[Select[Rest@ #2, OddQ], PrimeQ], !IntegerQ@ Log2@ #1] & @@ {#, NestWhileList[If[EvenQ@ #, #/2, 3 # + 1] &, #, # > 2 &, 1, Infinity, -1]} &] (* Michael De Vlieger, Nov 07 2018 *)