TITLE: Perfect numbers and perfect powers
QUESTION [14 upvotes]: This was asked  earlier  at MSE.
The observation that 28 = 27 + 1 shows that it is possible to have consecutive perfect numbers and perfect powers.  However, this must be extremely rare.  Is it unique?
Questions: (1) Is there another example known of such a consecutive pair?
(2)  Is there a rigorous or heuristic proof showing that there are only finitely many such pairs?
Thanks

REPLY [2 votes]: Actually a stronger result (for cube powers) is proved here:   
Luis H. Gallardo, "On a remark of Makowski about perfect numbers"
Elem. Math. 65 (2010) 121 – 126 

The only even perfect number that is also a sum of two cubes is 28.

The paper is easy (and free) to read.  I believe you will enjoy it!