A000079 and the least deficient numbers

[hv at crypt.org]

Max <relf at unn.ac.ru> wrote:
:
:A000079 claims that the powers of 2 coincides with the least deficient numbers. This claim is repeated by MathWorld at http://mathworld.wolfram.com/LeastDeficientNumber.html with the only reference to A000079.
:
:Is it trivial to show that there are no other least deficient numbers besides the powers of 2?
:Do you have any references to the proof?

I'm not sure how trivial it is; in general, you need sigma(n) odd, and
therefore n = 2^k m^2 for some odd m. That then leads you to:
  (sigma(m^2) - 1) / (sigma(m^2) - m^2) = 2^(k+1)

Now when m = p^q that requires p = 2^(k+1), an obvious contradiction; I'm
not sure whether there is another obvious contradiction to rule out the
cases where m is the product of 2 or more primes.

Hugo