Just for grins, I created lists of first-entries of finite sequences of rank $r$ for the Syracuse problem (Collatz conjecture using only odd numbers) and found these sequences on OEIS. My sequences, constructed by the expression on RHS below, are odd numbers only and are the same as the odd numbers in each $\zeta(k)-1$ sequence.
With $k=2,$ A143028
With $k=3,$ A143029
With $k=4,$ A143030 (odd numbers only) $\cong$ With $r=k-1, \{(2 c-1) 2^r-1\}_{c=1}^{8}$
$\{ 7, 23, 39, 50, 55, 71, 87, 103, 104, 119\}$ (odd numbers only) $\cong$
$ \{ 7, 23, 39, { }\uparrow{ }, 55, 71, 87, 103,{ }{ }{ }\uparrow{ }{ }{ }, 119\}$
$\uparrow$ points to an interstitial even number.
With $k=5,$ A143031
With $k=6,$ A143032
With $k=7,$ A143033
With $k=8,$ A143034
With $k=9,$ A143035
With $k=10,$ A143036
These OEIS sequences are from:
William J. Keith Sequences of density zeta(k) - 1
Would this indicate a connection between the Collatz conjecture and densities in the $\zeta(k)-1$ sequence?
Edit The connection is coincidental. However, these sequences can be produced by a simple sieving method:
Start with a list of all odd numbers. Remove every other number and save to a list. This will be the $\zeta(k)-1$ odd numbers. Repeat this step for each successive $k$.
The count of odd numbers is halved at each step.