A000079 and the least deficient numbers

[Jack Brennen]

 > 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 haven't seen any proof of this.  It seems to be an open problem.
See:  http://mathworld.wolfram.com/AlmostPerfectNumber.html

In particular, "The only *known* almost perfect numbers are the powers of 2."

(My emphasis.)



Heuristics would seem to indicate that there is a very good probability
that there are no other "almost perfect" or "least deficient" numbers,
because assuming any reasonable distribution of sigma(), the "probability"
that sigma(X) is equal to 2*X-1 will be proportional to 1/X.  Since the
only numbers for which this is possible are of the form N^2 or 2*N^2,
the total number of "randomly" occurring solutions can be guessed to be
proportional to the sum over all positive N of 1/N^2+1/(2*N^2).  Said
sum converges to the value Pi^2/4, or about 2.4674.  Considering that
no other such numbers have been found to a fairly high search height
(and that the sum of 1/N^2 converges very quickly), it seems reasonable
to believe that none will ever be found.