# 1, 4, 8, 9, 16, 25, 27, 32

**Number Theory**Level 3

Find the largest positive integer \(n<100,\) such that there exists an arithmetic progression of positive integers \(a_1,a_2,...,a_n\) with the following properties.

1) All numbers \(a_2,a_3,...,a_{n-1}\) are powers of positive integers, that is numbers of the form \(j^k,\) where \(j\geq 1\) and \(k\geq 2\) are integers.

2) The numbers \(a_1\) and \(a_{n}\) are not powers of positive integers.