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

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


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

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.

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What is T(n,k)? –  frogeyedpeas Jan 1 at 8:40
@frogeyedpeas It usually denotes the number of elements in the set. –  karvens Jan 1 at 8:47
For which values of $k$ have you verified that $S(k)=T(k)$? –  Gerry Myerson Jan 2 at 3:58
oeis.org/A126959 has been calculated out to $n=36$, so you have verified $S(k)=T(K)$ out to $k=36$? –  Gerry Myerson Jan 2 at 4:07
@GerryMyerson, One sequence fails at $k=10$ and the other at $k=17$. Yikes! –  Fred Kline Jan 2 at 4:34

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