Odd Perfect Numbers?

Is there some characterization of the sum of divisors for an odd square number?

 

The sum-of-divisors is multiplicative, and sigma(p^k) = 1 + p + \cdots
+ p^k = (p^{k+1}-1}/(p-1).

So the sum of divisiors of an odd square number is always of the form

(p_1^{2k_1+1}-1)(p_2^{2k_2+1} -1) * ... * (p_m^(2k_m+1))/(p_1-1)
(p_2-1)...(p_m-1)

or

(1+p_1+...+p_1^{2k_1})(1 + p_2 + ... + p_2^{2k_2})...(1 + p_m + ... +
p_m^{2k_m}).

with p_1<p_2<...<p_m odd primes, k_1,k_2,...,k_m positive integers.
And any such number is the sum of the divisors of an odd square.