$\sigma$ and $\tau$ are the sum of divisors and divisor counting functions respectively. In general it does not seem to be the case that for any $n$, $\sigma(\tau(n))=\tau(\sigma(n))$ so it seems peculiar that this is true for the consecutive numbers 170, 171, and 172.
I was aware previously that 170 is the smallest number such that $\varphi(170)$ and $\sigma(170)$ are perfect squares.
Since the $\sigma$ function appears in both, I was wondering if one of these facts can be derived from the other.
Specifically, does 170 being the smallest number that makes $\varphi(170)$ and $\sigma(170)$ perfect squares enough to imply that 170, 171, and 172 must result in $\sigma(\tau(n))=\tau(\sigma(n))$? (or the other way around)