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Perfect Power


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.
(1)

As shown by Goldbach, the sum of reciprocals of perfect powers (excluding 1) with duplications converges,

 sum_(m=2)^inftysum_(k=2)^infty1/(m^k)=1.
(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...
(3)

(OEIS A072102), where mu(k) is the Möbius function and zeta(k) is the Riemann zeta function.

The numbers of perfect powers without duplications less than 10, 10^2, 10^3, ... are 4, 13, 41, 125, 367, ... (OEIS A070428).


See also

Achilles Number, Niven's Constant, Odd Power, Power

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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."

Referenced on Wolfram|Alpha

Perfect Power

Cite this as:

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

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