ÖMO 2009 Problem 5

Let $n>1$ and for $1 \leq k \leq n$ let $p_k = p_k(a_1, a_2, . . . , a_n)$ be the sum of the products of all possible combinations of k of the numbers $a_1,a_2,...,a_n$ . Furthermore let $P = P(a_1, a_2, . . . , a_n)$ be the sum of all $p_k$ with odd values of $k$ less than or equal to $n$ .

How many different values are taken by $a_j$ if all the numbers $a_j (1 \leq j \leq n)$ and $P$ are prime?