The divisor product of a positive integer 
is
 |
(1)
|
where the product is over the positive divisors 
of 
(including 
itself). For 
, 2, ..., the first few values
are 1, 2, 3, 8, 5, 36, 7, 64, 27, 100, 11, 1728, 13, 196, ... (OEIS A007955).
The divisor product satisfies the identity
 |
(2)
|
The following table gives values of 
for which 
is a 
th power. Lionnet (1879) considered
the case 
.
 | OEIS |  |
| 2 | A048943 | 1, 6, 8, 10, 14, 15, 16, 21, 22, 24, 26, ... |
| 3 | A048944 | 1, 4, 8, 9, 12, 18, 20, 25, 27, 28, 32, ... |
| 4 | A048945 | 1, 24, 30, 40, 42, 54, 56, 66, 70, 78, ... |
| 5 | A048946 | 1, 16, 32, 48, 80, 81, 112, 144, 162, ... |
Write the prime factorization of a number 
,
 |
(3)
|
Then the power of 
occurring in 
is
 |
(4)
|
(Kaplansky 1999). For 
, this allows rules for determining
when 
is a power of 
to be determined, as considered by Halcke (1719) and Lionnet
(1879). Let 
, 
, and 
be distinct primes, and the
following table gives the conditions and first few 
for which 
is a given power 
of 
(Ireland and Rosen 1990, Kaplansky 1999, Dickson 2005). The
case of third powers corresponds to numbers having exactly
six divisors, the case of fourth powers
to numbers having eight divisors, and so on.
 | forms | Sloane |  |
| 2 |  ,  | A007422 | 6, 8, 10, 14, 15, 21, 22, ... |
| 3 |  ,  | A030515 | 12, 18, 20, 28, 32, 44, ... |
| 4 |  ,  ,  | A030626 | 24, 30, 40, 42, 54, 56, ... |
| 5 |  ,  | A030628 | 48, 80, 112, 162, 176, ... |
The unit value 
satisfies 
for every positive integer 
. It is omitted from the second table, including the
rows for OEIS A007422 and OEIS A030628,
whose sequences include it.
