Let $N$ be a positive integer near $$3^{3^{3^3}}$$ Suppose , it has no prime factor below $10^{11}$ as it is the case for $$3^{3^{3^3}}+2^{2^{2^2}}$$
The random variable $X$ denotes the number of digits of the smallest prime factor of $N$. Per assumption , we know $X\ge 12$
What is the smallest positive integer $k$ such that $P(X\le k)>0.99$ holds ?
In other words , what is the largest number of digits $k$ such we can safely assume that the smallest prime factor has at most $k$ digits ?
I tried to use that the a-priori probability that there is no prime factor below $n$ is $\frac{e^{-\gamma}}{\ln(n)}$ and that the sum of the reciprocals of the primes in the range $[a,b]$ is about $\ln(\frac{\ln(b)}{\ln(a)})$ , but I could not derive a formula for the probability $P(X\le k)$