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Prime Power


A prime power is a prime or integer power of a prime. A test for a number n being a prime is implemented in the Wolfram Language as PrimePowerQ[n].

The first few prime powers are 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, ... (OEIS A000961). The first few prime powers with power >=2 are given by 4, 8, 9, 16, 25, 27, 32, 49, 64, 81, ... (OEIS A025475). The number of prime powers (exponents >=2) up to x is given by

 x^(1/2)+x^(1/3)+x^(1/4)+...=O(x^(1/2)lnx)

(Hardy 1999, p. 27).

The following table gives prime kth powers.

kOEISprime kth powers
1A0000402, 3, 5, 7, 11, 13, 17, 19, 23, ...
2A0012484, 9, 25, 49, 121, 169, 289, 361, ...
3A0300788, 27, 125, 343, 1331, 2197, 4913, ...
4A03051416, 81, 625, 2401, 14641, 28561, 83521, ...
5A05099732, 243, 3125, 16807, 161051, 371293, ...

See also

Prime Number, Solitary Number

Explore with Wolfram|Alpha

References

Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, 1999.Sloane, N. J. A. Sequences A000040/M0652, A000961/M0517, A001248, A025475, A030078, A030514, and A050997 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Prime Power

Cite this as:

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

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