Conjecture---Identity for Sieve of Eratosthenes collisions.

Question

Let

$$\beta(n,k) = \max_{d \leq k}(d|n)$$

$$S(k)= \sum_{n=1}^{k!} \beta(n,k),$$

$\hspace{20mm}$and

$$T(k)=\# \{ ~i\cdot j~~\big|_{i=1}^k \big|_{j=1}^{k!} \}$$

Does $$S(k)=T(k)?$$

See OEIS A126959.
Replace $k!$ in $S,T$ with $\exp (\psi(k) )$, where $\psi(\cdot)$ is second Chebyshev function, to get A101459.