Problems & Puzzles: Puzzles

Puzzle 542.- Sigma (x)=q^n

Farideh Firoozbakht sent the following puzzle:

Sigma(3^4) = 11^2
Sigma(20^2) = 31^2.

Q1. Can you find other numbers n such that sigma(n) = q^k  where q is an odd prime and k is greater than 1 ?

Q2. Can you find another prime p such that sigma(p^m) = q^n where m & n are integers greater than 1 and q is prime ?

_______________
Note that for Mersenne primes p sigma(p) is of the form 2^k.