How do we solve a tight big-O bound for the recurrence $T(n) = T(n^{2/3}) + 1$?

Question

The big-O bound seems largely governed by how many times we can take the input $n$ by the $\frac{2}{3}$ power until it reaches some constant like 1.

How do I start formalizing this problem in math terms?

Answer 1

Hint: Let $a=\frac32$. First find a recursion for $$U(k)=T(\exp(a^k)),$$ then deduce a formula for $U(k)$, finally determine for which $k$, $$\exp(a^k)\approx n, $$ when $n$ is large.