Vasya's Arithmetic 2

After learning about the factorial in programming class, Vasya's interest in mathematics grew significantly. Wanting to contribute something new, he invented a new operation called the multifactorial and began studying it in detail.

First, Vasya introduced the concept of the simple factorial. According to Vasya, the simple factorial of a given number $n$ is the product of all positive integers, starting from $n$ in descending order, where each factor is one less than the previous one.

Naturally, Vasya then introduced the concepts of the $2$-factorial, $3$-factorial, and, in general, the $k$-factorial, which he called the multifactorial of order $k$.

The multifactorial of order $k$, according to Vasya, is the product of all positive integers in descending order, starting from $n$, where each factor is $k$ less than the previous one.

Vasya illustrated these concepts with the following formulas:

n! = n \cdot (n-1) \cdot (n-2) \cdot (n-3) \cdot \ldots

n!! = n \cdot (n-2) \cdot (n-4) \cdot (n-6) \cdot \ldots

n!!! = n \cdot (n-3) \cdot (n-6) \cdot (n-9) \cdot \ldots

In general, the formula can be written as:

Inspired to tie this new branch of mathematics to real life, Vasya wondered: how many different divisors does the multifactorial of a given order $k$ have?