# Conjecture---Identity for Sieve of Eratosthenes collisions.

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.

• What is T(n,k)? Jan 1 '15 at 8:40
• @frogeyedpeas It usually denotes the number of elements in the set. Jan 1 '15 at 8:47
• For which values of $k$ have you verified that $S(k)=T(k)$? Jan 2 '15 at 3:58
• oeis.org/A126959 has been calculated out to $n=36$, so you have verified $S(k)=T(K)$ out to $k=36$? Jan 2 '15 at 4:07
• @GerryMyerson, One sequence fails at $k=10$ and the other at $k=17$. Yikes! Jan 2 '15 at 4:34