Antti Karttunen: By taking the inverse of A122111v4 given at the first PinkBox-comment Sun Jan 04 16:52, we get yet another "entanglement-recurrence" for A122111 (as it is self-inverse): 
(definec (A122111v5 n)
   (cond ((= n 1) n)
         ((zero? (A241917 n)) (A003961 (A122111v5 (A052126 n))))
         (else (* 2 (A122111v5 (A252462 n))))
   )
)

Antti Karttunen: We can represent A122111 as a binary tree ("entanglement") permutation, weird!

Antti Karttunen: And I assume the same holds for A241909 (see the first PinkBox-comment in https://oeis.org/draft/A253563).
Now, the next big question: can we represent A243505(n) (=  A122111(A064216(n))) in the same way? Because then that would give us a key to represent both A064216 and its inverse A048673 in similar way, my goal from a few months ago.

Antti Karttunen: Similarly, A250245/A250246 can be represented so, see e.g. https://oeis.org/draft/A252753 - https://oeis.org/draft/A252756

Antti Karttunen: Of course the fundamental question is: Can ANY permutation of natural numbers represented as an "entanglement permutation" of two pairs of complementary sets of natural numbers, A/B & C/D ? (Note that some of A, B, C & D could be identical with each other).

Antti Karttunen: And this started last night when I reclined in my bed and suddenly realized the "double-Ω formula": A001222(n) + A001222(A122111(n)) = A001222(n) + A061395(n), from which follows that A122111 preserves A252464:
(same-intfuns1? A252464 (COMPOSE A252464 A122111) 1024) --> #t
so permutations like https://oeis.org/draft/A253563 can be represented as  a(n) = A122111(A005940(n+1)).

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Antti Karttunen: (same-intfuns1? A122111 (COMPOSE A253565 A243071) 256) --> #t
AND:
(define (A122111v4 n)
  (cond ((<= n 2) n)
        ((even? n) (A253550 (A122111v4 (/ n 2))))
        (else (A253560 (A122111v4 (A064989 n))))
  )
)

AND (same-intfuns1? A122111 A122111v4 512) --> #t