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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.
5
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
1,2
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
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
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
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