Odd abundant number question

[hv at crypt.org]

hv at crypt.org wrote:
:"David Wilson" <davidwwilson at comcast.net> wrote:
::Is an odd abundant number necessarily a sum of proper divisors of itself?
:
:Checking UPINT, I find that a number is called "pseudoperfect" if it is
:the sum of some of its proper divisors, and "weird" if it is abundant
:but not pseudoperfect [...]
:
:However I can see no particular reason why such a thing shouldn't exist -
:it is merely the sparsity of small abundant odd numbers that makes it
:difficult to check. An abundant number is weird or not precisely if the
:abundance itself can be expressed as a sum of the divisors [...]

Hmm, now I'm not so sure. Note that the abundance of n is (sigma(n) - 2n).
Now given m pseudoperfect, n prime to m:
  sigma(mn) - 2mn = sigma(m) . sigma(n) - 2mn
      = n . (sigma(m) - 2m) + sigma(m) . (sigma(n) - n)

Since m is pseudoperfect, (sigma(m) - 2m) is expressible as a sum of
the proper divisors of m, and (sigma(n) - n) is precisely the sum of
the proper divisors of n, so mn is also necessarily pseudoperfect (and
so not weird).

The search for an odd weird number can therefore ignore all but "primitive"
abundant numbers - ie ignore any mn where m is abundant and (m, n) = 1.
That'll help reduce the search space, but also makes me less confident
of my original assertion that one must exist.

Hugo