

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.


2



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
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OFFSET

1,2


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.


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. 194201.
Eric Weisstein's World of Mathematics, Collatz Problem
Wikipedia, Collatz conjecture
Index entries for sequences related to 3x+1 (or Collatz) problem


EXAMPLE

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.


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 *)


PROG

(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


CROSSREFS

Cf. A014682, A061641, A127633.
Sequence in context: A224694 A140900 A166936 * A276623 A121519 A098968
Adjacent sequences: A166242 A166243 A166244 * A166246 A166247 A166248


KEYWORD

nonn


AUTHOR

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


EXTENSIONS

Edited by Ralf Stephan, Nov 26 2013
Definition clarified by N. J. A. Sloane, Mar 17 2017


STATUS

approved



