Perfect power

In mathematics, a perfect power is a positive integer that can be resolved into equal factors, and whose root can be exactly extracted. i.e., a positive integer that can be expressed as an integer power of another positive integer. More formally, n is a perfect power if there exist natural numbers m > 1, and k > 1 such that mk = n. In this case, n may be called a perfect kth power. If k = 2 or k = 3, then n is called a perfect square or perfect cube, respectively. Sometimes 0 and 1 are also considered perfect powers (0k = 0 for any k > 0, 1k = 1 for any k). In mathematics, a perfect power is a positive integer that can be resolved into equal factors, and whose root can be exactly extracted. i.e., a positive integer that can be expressed as an integer power of another positive integer. More formally, n is a perfect power if there exist natural numbers m > 1, and k > 1 such that mk = n. In this case, n may be called a perfect kth power. If k = 2 or k = 3, then n is called a perfect square or perfect cube, respectively. Sometimes 0 and 1 are also considered perfect powers (0k = 0 for any k > 0, 1k = 1 for any k). A sequence of perfect powers can be generated by iterating through the possible values for m and k. The first few ascending perfect powers in numerical order (showing duplicate powers) are (sequence A072103 in the OEIS): The sum of the reciprocals of the perfect powers (including duplicates such as 34 and 92, both of which equal 81) is 1: