A166245 Numbers n such that the Collatz trajectory of n (iterate T(k)=k/2 if k is even, (3k+1)/2 if k is odd, A014682, starting at n and stopping if you reach 1) never exceeds n.

1, 2, 4, 8, 10, 12, 16, 20, 24, 26, 28, 32, 34, 36, 40, 42, 44, 48, 50, 52, 56, 58, 64, 66, 68, 72, 74, 76, 80, 84, 88, 90, 92, 96, 98, 100, 104, 106, 112, 114, 116, 120, 122, 128, 130, 132, 136, 138, 140, 144, 148, 152, 154, 156, 160, 162, 168, 170, 172, 176, 178, 180

Offset: 1

Author

Michael Higgins (mikehiggins1981(AT)gmail.com), Oct 10 2009

Keywords

• nonn

Comments

Let T(n)=n/2 if n is even, (3n+1)/2 if n is odd. This function is the same as the one in the Collatz conjecture, 3x+1 problem, Kakutani's Problem, Syracuse problem etc. Then x is an element of the sequence iff T^k(x) <= x for all k. Several conjectures relating to the 3x+1 problem can be restated in terms of this set. For example: There are no nontrivial cycles iff the <= can be replaced with < in the definition of the sequence for x>2. x has bounded trajectory iff T^k(x) is an element of the sequence for some k. These two statements together are equivalent to the Collatz conjecture.

Examples

			1 is a term, because the trajectory stops right there at 1.
2 is a term because the trajectory is 2->1.
3 is not a term because the trajectory is 3 -> 5 -> 8 -> 4 -> 2 -> 1, and 5>3.
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000
• Douglas J. Shaw, The Pure Numbers Generated by the Collatz Sequence, The Fibonacci Quarterly, Vol. 44, Number 3, August 2006, pp. 194-201.
• Eric Weisstein's World of Mathematics, Collatz Problem
• Wikipedia, Collatz conjecture
• Index entries for sequences related to 3x+1 (or Collatz) problem

Crossrefs

Cf. A014682, A061641, A127633.

Programs

Mathematica

L1 = {}; For[i = 1, i < 4096, i++, max = i; n = i; While[n != 1 || Element[n, L1] == False, If[Mod[n, 2] == 1, n = (3 n + 1)/2; If[max <= n, max = n], n = n/2; If[max <= n, max = n]]]; Sort[DeleteDuplicates[L1]]];
ctenQ[n_]:=Max[NestWhileList[If[EvenQ[#],#/2,(3#+1)/2]&,n,#>1&]]<=n; Select[Range[200],ctenQ] (* Harvey P. Dale, Mar 17 2017 *)

PARI

is(x)=my(X);X=x;while(x!=1,x=if(x%2,(3*x+1)/2,x/2);if(x>X,return(0)));1

Extensions

Edited by Ralf Stephan, Nov 26 2013

Definition clarified by N. J. A. Sloane, Mar 17 2017