IMC 2013 Problem 5

Does there exist a sequence ${\left( {{a_n}} \right)}$ of complex numbers such that for every positive integer ${p}$ we have that ${\sum\limits_{n = 1}^{ + \infty } {a_n^p} }$ converges if and only if ${p}$ is not a prime?

Proposed by Tomáš Bárta, Charles University, Prague.