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

%I A177000 #27 Feb 16 2025 08:33:12
%S A177000 2,3,5,7,11,13,17,19,29,37,53,59,67,89,101,131,149,157,179,181,197,
%T A177000 241,269,277,349,397,739,853,1109,1237,1429,1621,1861,1877,2161,2389,
%U A177000 2531,2957,3413,3797,4549,5717,7621,10069,13397,17749,20021,31541,40277
%N A177000 The Collatz iteration of these primes produces only even numbers, primes and 1.
%C A177000 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.
%C A177000 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
%H A177000 Donovan Johnson and T. D. Noe, <a href="/A177000/b177000.txt">Table of n, a(n) for n = 1..10000</a> (Donovan Johnson to 203 terms)
%H A177000 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CollatzProblem.html">Collatz Problem</a>
%H A177000 Wikipedia, <a href="http://en.wikipedia.org/wiki/Collatz_conjecture">Collatz conjecture</a>
%H A177000 <a href="/index/3#3x1">Index entries for sequences related to 3x+1 (or Collatz) problem</a>
%e A177000 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.
%t A177000 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]]
%t A177000 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 *)
%o A177000 (Haskell)
%o A177000 a177000 n = a177000_list !! (n-1)
%o A177000 a177000_list = filter (all (\x -> even x || a010051' x == 1) .
%o A177000                        (init . a070165_row)) a000040_list
%o A177000 -- _Reinhard Zumkeller_, Apr 03 2012
%o A177000 (PARI) is(n)=isprime(n) && (n<23 || is((3*n+1)>>valuation(3*n+1,2))) \\ _Charles R Greathouse IV_, Jun 20 2013
%Y A177000 Cf. A055509, A078350, A078373, A070165, A010051.
%K A177000 nonn,nice
%O A177000 1,1
%A A177000 _T. D. Noe_, Apr 30 2010