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%I #21 Mar 17 2017 10:55:59
%S 1,2,4,8,10,12,16,20,24,26,28,32,34,36,40,42,44,48,50,52,56,58,64,66,
%T 68,72,74,76,80,84,88,90,92,96,98,100,104,106,112,114,116,120,122,128,
%U 130,132,136,138,140,144,148,152,154,156,160,162,168,170,172,176,178,180
%N 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.
%C 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.
%H Harvey P. Dale, <a href="/A166245/b166245.txt">Table of n, a(n) for n = 1..1000</a>
%H Douglas J. Shaw, <a href="http://www.fq.math.ca/Papers1/44-3/quartshaw03_2006.pdf">The Pure Numbers Generated by the Collatz Sequence</a>, The Fibonacci Quarterly, Vol. 44, Number 3, August 2006, pp. 194-201.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CollatzProblem.html">Collatz Problem</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Collatz_conjecture">Collatz conjecture</a>
%H <a href="/index/3#3x1">Index entries for sequences related to 3x+1 (or Collatz) problem</a>
%e 1 is a term, because the trajectory stops right there at 1.
%e 2 is a term because the trajectory is 2->1.
%e 3 is not a term because the trajectory is 3 -> 5 -> 8 -> 4 -> 2 -> 1, and 5>3.
%t 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]]
%t ctenQ[n_]:=Max[NestWhileList[If[EvenQ[#],#/2,(3#+1)/2]&,n,#>1&]]<=n; Select[Range[200],ctenQ] (* _Harvey P. Dale_, Mar 17 2017 *)
%o (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
%Y Cf. A014682, A061641, A127633.
%K nonn
%O 1,2
%A Michael Higgins (mikehiggins1981(AT)gmail.com), Oct 10 2009
%E Edited by _Ralf Stephan_, Nov 26 2013
%E Definition clarified by _N. J. A. Sloane_, Mar 17 2017
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