Pensieve header: Collatz with $k\to k/3$ or $(5k\pm 1)/3$.

Clear[vis]; n=0;
While[n≤1000,
path={k=++n};
While[!TrueQ[vis[k]],
vis[k]=True;
k1=Switch[Mod[k,3], 0, k/3, 1, (5k+1)/3, 2, (5k-1)/3];
If[TrueQ[vis[k1]], Break[], AppendTo[path,k=k1]]
];
If[Length[path]>1, Echo[path]]
]