Let be N the integer from which you start the Collatz sequence.
And let be Mx(N) the maximum reached starting from N.
Skipping trivial trajectories, are there infinitely many N such that:
$N\equiv 3^j\pmod {42}$ and $Mx(N) \equiv 2^k\pmod {42}$, for some positive k and j?
N=993 and Mx(993)=8080 are an example.
Infact 993 $\equiv 27\pmod {42}$ and 8080 $\equiv 16\pmod {42}$
N=27 and Mx(27)=9232 are a non example because 9232 is not congruent to $2^k\pmod {42}$
Is N=993 and Mx(993) the least example? Or are there N's<993 with such property?
$N=3$ and $Mx(3)=16$ satisfy your requirements. For more example check this code here