Hyper Harmonic number

Harmonic numbers are almost the most harmonious. However, the ordinary harmonic series does not show the same beauty and simplicity as the sequence invented by the rabbit Brian. Define the $n$-th harmonic number as follows:

Brian believes that such a harmonic number is still not the most harmonious. He believes that there is a hyper harmonic number, and defines it as:

= H_1 \cdot H_2 \cdot H_3 \cdot ... \cdot H_k,

i.e.

Brian believes that the number will become even more harmonic if it is calculated modulo $n$, where $n$ is a prime number.

While investigating hyper harmonic numbers, Brian noticed that starting from some k_z, all subsequent hyper harmonic numbers are equal to zero (calculated modulo $n$). Brian called the number k_z the hyper harmonic dimension of $n$.

Formally, k_z is such that for all 1 \leq k < k_z, the hyper harmonic number is not $0$ modulo $n$, and for k_z \leq k \leq n - 1, it is $0$ modulo $n$ (all calculations are done modulo $n$).

Find the hyper harmonic dimension for the given prime $n$.