I believe that consistent balance provided by persistent structural support is the key to all of Number Theory’s perplexities. The number line is really nothing more than an architectural structure, and the maintenance of balance about a midpoint is key to structural stability. Essentially, Number Theory is just a mental game of Jenga.
Even perfect number demonstrates balance when you tack on improper divisors –
1, 2, 4, 8, 16, 32, |64, 127,| 254, 508, 1016, 2032, 4064, 8128
Note the centrality of divisors 2^k and ((2^k+1)-1)
There is a central dividing line between a finite sequence of powers of two and a finite sequence of doublings beginning with an odd prime. (Of course, the powers of two are similarly a sequence of doublings beginning with the unit integer 1.)
Two sequential sets with an equal number of elements – the makings of one-to-one correspondence.
What sort of balance or structural integrity could possibly be demonstrated amongst the divisors of an odd perfect number?
In a Cantorian sense, I don't hold much hope for an odd perfect number.