Landau Function finding Runtime recursive of irrational function.

Question

I have this question: what is the runtime (Theta) of this recursive function:
$T(n) = T(n^{\frac{2}{3}}) + 20$
The problem is, I don't know what to do, the recursive function gets to a constant after infinity iterations which makes the problem hard. The master's theorem does not work either as we can't use it (it's not linear aT(n/b) + f(n) ..)
Any help would be appreciated.

Answer 1

I think i have found the answer.
$n^{(\frac{2}{3})^t} = 2$ we get: $k = \log_{1.5}(log_2(n))$ and we prove that this is Theta of $\log(\log(n))$ by choosing c1 and c2 such that
$c_1 \cdot \log(\log(n)) \leq \log_{1.5}(log_2(n)) \leq c_2 \cdot \log(\log(n))$

Answer 2

I do not know any computer-science, but have you tried letting $T(n)=S(\log(n))$ so that $$S(n)=S\left(\frac{2}{3}n\right)+20$$ On which you can apply the Master theorem and then revert $S$ back to $T$? I do not know if this is allowed.