# Divisors of a $k$-multiperfect number

Let $$n$$ be a $$k$$-multiperfect number. Denote by $$d_m$$ its $$m$$ smallest divisor, and $$n_{m}$$ the number of divisors of $$n$$ divisible by $$d_m$$. Is there for all $$2\leq m\leq\tau(n)$$ an integer $$s(m)$$ such that $$n_m=d_{s(m)}$$? Does the function $$s$$ admit a fixed point ?

Edit : actually $$n_{m}=\tau(n/d_{m})$$. A sufficient condition for a number $$n$$ to fulfill $$d=\tau(n/d)$$ is to take $$n=\prod_{r}p_{r}^{p_{\sigma(r)}-1}$$ with $$\sigma$$ a permutation of the set of primes $$p$$ such that $$v_{p}(n)\geq 1$$ and $$d=\prod_{p, v_{p}(n)\geq 1}p$$ .