denote the product of the 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 .
(Kaplansky 1999). 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, then 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 forth powers to numbers
having eight divisors, and so on.
, let
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).
for which 
is a 
th power. Lionnet (1879) considered the case 
.
,
occurring in 
is
is a power of 
to be determined, as considered by Halcke (1719) and Lionnet
(1879). Let 
,
, and 
be distinct primes, then 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 forth powers to numbers
having eight divisors, and so on.
, 
,
,
, 
,
