Problem of the Day #20

POTD December 9, 2023
Author: munch

Find the number of values of n, such that n \in \mathbb{N} and n > 1, for which there exists a set of positive integers \{a_1, a_2, ..., a_{n-1}\}, which satisfy the following conditions:

1. For 1 \leq i \leq n-1, a_i > i and n \nmid a_i
2. For 1 \leq i < j \leq n-1, a_i \text{ is not} \equiv a_j (\bmod n).
3. There exists k, such that k \in \mathbb{Z}^+, k > 1 and 1+k^{a_1}+k^{a_2}+...+k^{a_{n-1}} is prime.