A perfect power is a number n of the form m^k, where m>1 is a positive integer and k>=2. If the prime factorization of n is n = p_1^(a_1) p_2^(a_2) ...p_k^(a_k), then n is a perfect power iff GCD(a_1, a_2, ..., a_k)>1. Including duplications (i.e., taking all numbers up to some cutoff and taking all their powers) and taking m>1, the first few are 4, 8, 9, 16, 16, 25, 27, 32, 36, 49, 64, 64, 64, ... (OEIS A072103). Here, 16 is duplicated since 16 = 2^4 = 4^2. As shown by Goldbach, the sum of reciprocals of perfect powers (excluding 1) with duplications converges, sum_(m = 2)^∞ sum_(k = 2)^∞ 1/m^k = 1.

Achilles number | Niven's constant | odd power | power