Is there a correlation between numbers with record totient valence and the factorials? For example, there are 10 values of $n$ such that $\phi(n) = 24$, and that's more than for any smaller, positive integer. It's not true of 120 but it is true of 720. I haven't verified it for 5040.
 A: There is a correlation but it's not as interesting as you might expect.
As you already know, $n! = \prod_{i = 1}^n i$. You also already know that the totient function is multiplicative conditioned on coprimality.
So, given a prime $p$ and a positive integer exponent $\alpha$, we have $\phi(p^\alpha) = p^{\alpha - 1}(p - 1)$. In order for an integer to have a totient valence, it must either be a number of that form or the product of numbers of that form. Some integers can't be represented that way at all, they're "nontotients." Other integers can be represented that way in more than one way, like 24, for which we have $(5 - 1)(7 - 1) = (3 - 1)(13 - 1) = (5 - 1)(3^{2 - 1}(3 - 1)) =$ etc.
Clearly $n!$ has numbers that are one less than primes and powers of primes among its factors. But it also has nontotients among its factors, though these can contribute to the valence (e.g., $2 \times 14 = 29 - 1$. If $n$ is a nontotient, $n!$ is unlikely to set a record for totient valence, but otherwise, it might.
T. D. Noe and Donovan Johnson have computed a list of numbers with record-setting totient valence up to 3832012800, see http://oeis.org/A097942.
