A177000 The Collatz iteration of these primes produces only even numbers, primes and 1.

2, 3, 5, 7, 11, 13, 17, 19, 29, 37, 53, 59, 67, 89, 101, 131, 149, 157, 179, 181, 197, 241, 269, 277, 349, 397, 739, 853, 1109, 1237, 1429, 1621, 1861, 1877, 2161, 2389, 2531, 2957, 3413, 3797, 4549, 5717, 7621, 10069, 13397, 17749, 20021, 31541, 40277

Offset: 1

Author

T. D. Noe, Apr 30 2010

Keywords

• nonn
• nice

Comments

The Collatz iteration of primes of the form (10*4^k-1)/3 produces only one additional prime: 5. The Collatz iteration of primes of the form (13*4^k-1)/3 produces only two additional primes: 5 and 13. This sequence is probably infinite.

In a sense, these are the simplest Collatz iterations starting with a prime number. Except for the increases (3x+1) when an odd prime occurs, the sequence produced by starting with a(n) is decreasing. All the primes that occur in such a Collatz iteration are in this sequence. - T. D. Noe, Oct 05 2011

Examples

			The Collatz iteration of 7 produces 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, and 1, which are either even, prime, or 1.
		

Links

• Donovan Johnson and T. D. Noe, Table of n, a(n) for n = 1..10000 (Donovan Johnson to 203 terms)
• Eric Weisstein's World of Mathematics, Collatz Problem
• Wikipedia, Collatz conjecture
• Index entries for sequences related to 3x+1 (or Collatz) problem

Crossrefs

Cf. A055509, A078350, A078373, A070165, A010051.

Programs

Haskell

a177000 n = a177000_list !! (n-1)
a177000_list = filter (all (\x -> even x || a010051' x == 1) .
                       (init . a070165_row)) a000040_list
-- Reinhard Zumkeller, Apr 03 2012

Mathematica

Collatz[n_] := NestWhileList[If[EvenQ[ # ], #/2, 3#+1] &, n, #>1 &]; Reap[Do[p=Prime[n]; s=Most[Select[Collatz[p],OddQ]]; If[And@@PrimeQ[s], Sow[p]], {n,PrimePi[10^4]}]][[2,1]]
oenQ[n_]:=AllTrue[DeleteCases[Most[NestWhileList[If[EvenQ[#],#/2, 3#+1]&,n, #>1&]], ?PrimeQ],EvenQ]; Select[Prime[Range[5000]], oenQ] (* The program uses the AllTrue function from Mathematica version 10 *) (* _Harvey P. Dale, Dec 28 2014 *)

PARI

is(n)=isprime(n) && (n<23 || is((3*n+1)>>valuation(3*n+1,2))) \\ Charles R Greathouse IV, Jun 20 2013