A129304 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.

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, 36, 39, 40, 41, 43, 44, 47, 48, 49, 54, 56, 57, 62, 64, 65, 68, 71, 72, 73, 78, 80, 82, 86, 88, 91, 94, 96, 97, 98, 103, 105, 107, 108, 111, 112, 114, 121, 123, 124, 128, 129, 130

Offset: 1

Author

T. D. Noe, Apr 09 2007

Keywords

• nonn

Comments

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.

Examples

			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.
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..2000
• T. D. Noe, Plot of the 2000 possible halving/tripling pairs for n <= 7540196
• Index entries for sequences related to 3x+1 (or Collatz) problem

Crossrefs

Cf. A006666 (number of halving steps), A006667 (number of tripling steps).

Programs

Mathematica

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