User blog comment:MachineGunSuper/The Chained Naulz Function and The Abusing Naulz Function/@comment-32783837-20171223164232

This is a very basic form of recursion, similar to FGH-style recursion. Yes, your "Abusing Naulz" is much stronger than the simple naulz function, but not very strong--in fact, it's still primitive recursive.

The function N(n) = n ₵ ₵ ₵ ₵... ₵ ₵ ₵ ₵n with n " ₵"s is only roughly comparable to f4(n) in the fast-growing hierarchy.

Then the function И(n) = n Ҩ Ҩ Ҩ Ҩ... Ҩ Ҩ Ҩ Ҩn with n " Ҩ"s is only roughly comparable to  f 5 (n).

Then the function Ñ(n) = n₡₡₡₡...₡₡₡₡n with n " ₡"s is only roughly comparable to  f 6 (n).

Still mere finite ordinals.

So, why stop at the "Abusing Naulz"?

Why not generalize this to something like:

a₵1b = a₵b

a₵ n b =  a₵ n-1 ( a₵ n-1 ( a₵ n-1 ( a₵ n-1 (...( a₵ n-1 ( a₵ n-1 ( a₵ n-1 (b))))...)))) with  a₵ n-1 b " ₵ n-1 "s

Rules follow similarly to the ones you described for  ₵ ₵,  ₵ ₵ ₵, etc.

So, that would mean:

aҨb =  a₵ 2 b

a ₡b =  a₵ 3 b

a₵ 4 b iterates what you call "Abusing Naulz",  a₵ 5 b iterates  a₵ <sub style="font-weight:400;">4 b, etc.

Then, the function Nm(n) =  n ₵ ₵ ₵ ₵... ₵ ₵ ₵ ₵m n  with n " ₵"s is comparable to  f <sub style="font-weight:400;">m+4 (n), and Nn (n) is comparable to  f <sub style="font-weight:400;">ω (n).

Now THAT's an extension.