Wikipedia gives the definition of the Golomb-Dickman constant as $$\lambda=\lim_{n\rightarrow\infty}\frac1n\sum_{k=2}^n\frac{\log(P_1(k))}{\log(k)}$$Where $P_1(x)$ is the largest prime factor of $x$.
My question is if $$\lim_{k\rightarrow\infty}\frac{\log(P_1(k))}{\log(k)}$$converges or not, and if so to what number? The upper and lower bounds of $P_1(k)$ are $k$ and $2$ respectively. Taking the average of the two, we get $\frac{k+2}2$. Substituting this for $P_1(k)$, we get $1$. This is a heuristic argument, so I am not sure.
Edit: I am now suspecting that the limit in question doesn't exist because of the unpredictable nature of $P_1(k)$.
Credit to @TheSimpliFire:
The limit in question doesn't exist. This is because the limit is different for different subsequences (for example the primes and the powers of 2).