Perfect Power

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, ... (OEIS 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, ... (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

(3)

(OEIS 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, ... (OEIS A070428).

REFERENCES:

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."