I don't remember this stuff well, but
σ(n)=2n is the definition of a perfect number. It is known that if n is a perfect
even number it must take the form you describe (and 2^k - 1 must also be prime), but the question of existence of odd perfect numbers is
unsolved, so I don't think this is a fruitful direction to persue!
It looks to me it would be more fruitful to note that
τ(n) is odd iff n is a perfect square (and then show if n is a perfect square than
σ(n)=2n), although this seems to me unlikely to make use of multiplicativity (so probably not what's intended, given the hint).
Edit: A little more thought tells me the last paragraph does work, and does use multiplicativity.