Sequence of Consecutive Integers with Same Number of Divisors

Theorem
The following sequence of consecutive integers all have the same number of divisors, that is, $8$:
 * $40 \, 311, 40 \, 312, 40 \, 313, 40 \, 314, 40 \, 315$

This is the longest such sequence known.

Proof
In the below, $\sigma_0$ denotes the divisor counting function.

Then we have: