OFFSET
0,2
COMMENTS
Conjecture: k exists for all n.
In other words, given an integer n, there always exists at least an integer k and a pair of integers (a, b) such that n + k = 2^a/3^b where a is the number of halving steps to reach 1, and b is the number of tripling steps to reach 1, in the 3x+1 problem.
EXAMPLE
MATHEMATICA
lst={}; Do[Collatz[k_]:=NestWhileList[If[EvenQ[#], #/2, 3 #+1]&, k, #>1&]; nn=500; t={}; k=0; While[Length[t]<nn, k++; c=Collatz[k]; ev=Length[Select[c, EvenQ]]; od=Length[c]-ev-1; If[Floor[2^ev/3^od]-k==n, AppendTo[lst, k]; Break[]]], {n, 0, 60}]; lst
CROSSREFS
KEYWORD
nonn
AUTHOR
Michel Lagneau, Dec 01 2015
STATUS
approved