Vladimir Letsko asked: 3. Note that sigma(3^4) = 11^2. Does there exists another pair (p,r) such that p is prime, r > 1 and sigma(p^r) = q^s where q is prime and s > 1? Regarding question #3: sigma(3^4) = 11^2 is the only solution for p^r < 10^16. Donovan