A program can easily search for the answer to this question. It simply needs to test each odd number and see if it is equal to the sum of its divisors. If there is an odd perfect number, given enough time and memory, the computer will find it. On the other hand, if odd perfect numbers do not exist, the computer would simply work on, halting with no answer when it runs out of resources. Thus the problem is unsolvable.
Solvable problems belong to the group of languages called recursive.
For these problems the TM will return a definite answer yes or no in a finite period of time. Yet many of these cannot be practically solved because of limits of time or space. A particularly simple example of this is the traveling salesperson problem. There is a salesperson who has to visit 50 cities. He wishes to do so by traveling the least number of miles and by not visiting any city more than once.