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%I #12 Jul 04 2022 23:17:54
%S 1,2,3,4,5,6,7,8,9,10,11,12,14,16,17,18,20,22,24,25,27,28,31,32,33,34,
%T 36,39,40,41,43,44,47,48,49,54,56,57,62,64,65,68,71,72,73,78,80,82,86,
%U 88,91,94,96,97,98,103,105,107,108,111,112,114,121,123,124,128,129,130
%N Numbers whose Collatz trajectory does not have the same number of halving steps and the same number of tripling steps as that of any smaller number.
%C Note that if k is in this sequence, then 2k is also. The plot shows a very narrow triangle of the possible halving/tripling pairs. As n increases, the width of the triangle grows on its right edge.
%H T. D. Noe, <a href="/A129304/b129304.txt">Table of n, a(n) for n = 1..2000</a>
%H T. D. Noe, <a href="http://www.sspectra.com/math/A129304.gif">Plot of the 2000 possible halving/tripling pairs for n <= 7540196</a>
%H <a href="/index/3#3x1">Index entries for sequences related to 3x+1 (or Collatz) problem</a>
%e For each integer k, let the ordered pair (h,t) give the numbers of halving and tripling steps in the Collatz trajectory of k. The pairs for k = 1..16 are (0,0), (1,0), (5,2), (2,0), (4,1), (6,2), (11,5), (3,0), (13,6), (5,1), (10,4), (7,2), (7,2), (12,5), (12,5), (4,0). Thus 13 and 15 are not in this sequence because their pairs are the same as for 12 and 14, respectively.
%t Collatz[n_] := Module[{c1=0,c2=0,m=n}, While[m>1, If[EvenQ[m], c1++; m=m/2, c2++; m=3m+1]]; {c1,c2}]; s={}; t={}; n=0; While[Length[t]<100, n++; c=Collatz[n]; If[ !MemberQ[s,c], AppendTo[s,c]; AppendTo[t,n]]]; t
%Y Cf. A006666 (number of halving steps), A006667 (number of tripling steps).
%K nonn
%O 1,2
%A _T. D. Noe_, Apr 09 2007