For smallish values of m, like m=12, it is possible to enumerate all the integers n with , if you go about it systematically.
Remember that if is the factorisation of n into prime powers, then . So if you want to find the possible values of n with , you should start by looking for all the factors of m of the form .
Let's take the example m=12. The factors of 12 that are of that form are
The only ways that we can make 12 as a product of these numbers are , , , and .
Notice that there is no way to write 3 in the form , so we cannot make use of the factorisation . On the other hand, there are two ways to write each of the numbers 2 and 6 in that form, which gives us four ways to make use of the factorisation . Also, we can slip in a factor to any factorisation that does not already include a term arising from a power of 2, such as (but we can have, for example , where the 2 arises from rather than ).
Thus there are eight numbers n with , namely 13, 21, 26, 27, 28, 36, 42 and 54.