A perfect power is a number of the form , where is a positive integer and . If the prime factorization of is , then is a perfect power iff .

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, ... (Sloane's A072103). Here, 16 is duplicated since

(1)

As shown by Goldbach, 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, ... (Sloane's A117453).

The first few perfect powers without duplications are 1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 100, 125, 128, ... (Sloane's A001597). Even more amazingly, the sum of the reciprocals of these numbers (excluding 1) is given by

(3)

(Sloane's A072102), where is the Möbius function and is the Riemann zeta function.

The numbers of perfect powers without duplications less than 10, , , ... are 4, 13, 41, 125, 367, ... (Sloane's A070428).

Finch, S. R. "Niven's Constant." §2.6 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 112-115, 2003.

Gould, H. W. "Problem H-170." Fib. Quart. 8, 268, 1970.

Graham, R. L.; Knuth, D. E.; and Patashnik, O. "Binomial Coefficients." Ch. 5 in Concrete Mathematics: A Foundation for Computer Science, 2nd ed. Reading, MA: Addison-Wesley, p. 66, 1994.

Sloane, N. J. A. Sequences A001597/M3326, A070428, A072102, A072103, and A117453 in "The On-Line Encyclopedia of Integer Sequences."

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Weisstein, Eric W. "Perfect Power." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/PerfectPower.html