A perfect power is a positive integer 
of the form 
, where 
is a positive integer
and 
is an integer.
This convention includes 1. For 
, if the prime factorization
of 
is 
, then 
is a perfect power iff 
.
Each representation 
expresses a perfect power as an even power or an odd power according as 
is even or odd.
Some perfect powers, such as 
, are both.
Including duplications (i.e., taking all numbers up to some cutoff and taking all their powers) and taking 
,
the first few are 4, 8, 9, 16, 16, 25, 27, 32, 36, 49, 64, 64, 64, ... (OEIS A072103).
Here, 16 is duplicated since
 |
(1)
|
The sum of reciprocals of perfect powers (excluding 1) with duplications converges,
 |
(2)
|
The first few numbers that are perfect powers in more than one way are 16, 64, 81, 256, 512, 625, 729, 1024, 1296, 2401, 4096, ... (OEIS A117453).
The first few perfect powers without duplications are 1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 100, 121, 125, ... (OEIS A001597). Even more amazingly, the sum of the reciprocals of these numbers (excluding 1) is given by
![sum_(k=2)^inftymu(k)[1-zeta(k)] approx 0.874464368...](/images/equations/PerfectPower/NumberedEquation3.svg) |
(3)
|
(OEIS A072102), where 
is the Möbius function
and 
is the Riemann zeta function.
The numbers of perfect powers without duplications not exceeding 10, 
, 
, ... are 4, 13, 41, 125, 367, ... (OEIS A070428).
