I was just playing around in desmos and I think I found something that approximates Lambert W for whole numbers.
$$f(x)=a(\int_{0}^{1}(\sum_{n=1}^{x}t^n-1)dt)+b$$
Where $a\approx0.765424$ And $b\approx0.944602$
$a$ and $b$ were achieved through linearization, plugging $xe^x$ in for x.
Please point me in the right direction for proofing or disproving that as $x \to \infty$ $f(x) \to W(x)$
This is an interesting post.
In fact $$f(x)=\int_{0}^{1}\sum_{n=1}^{x}(t^n-1)dt=\psi (x+2)+\gamma -2$$
Looking at the empirical model $$W(x)=a + b f(x)$$ for $0 \leq x \leq 10^6$ by steps of $10$, we obtain with $(R^2=0.999875)$ $$\begin{array}{clclclclc} \text{} & \text{Estimate} & \text{Standard Error} & \text{Confidence Interval} \\ a & 0.174690 & 0.000366 & \{0.173973, 0.175408\} \\ b & 0.903778 & 0.000032 & \{0.903715, 0.903840\} \end{array}$$