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# Divisor Product

By analogy with the divisor function , let

 (1)

denote the product of the divisors of (including itself). For , 2, ..., the first few values are 1, 2, 3, 8, 5, 36, 7, 64, 27, 100, 11, 1728, 13, 196, ... (OEIS A007955).

The divisor product satisfies the identity

 (2)

The following table gives values of for which is a th power. Lionnet (1879) considered the case .

 OEIS 2 A048943 1, 6, 8, 10, 14, 15, 16, 21, 22, 24, 26, ... 3 A048944 1, 4, 8, 9, 12, 18, 20, 25, 27, 28, 32, ... 4 A048945 1, 24, 30, 40, 42, 54, 56, 66, 70, 78, ... 5 A048946 1, 16, 32, 48, 80, 81, 112, 144, 162, ...

Write the prime factorization of a number ,

 (3)

Then the power of occurring in is

 (4)

(Kaplansky 1999). This allows rules for determining when is a power of to be determined, as considered by Halcke (1719) and Lionnet (1879). Let , , and be distinct primes, then the following table gives the conditions and first few for which is a given power of (Ireland and Rosen 1990, Kaplansky 1999, Dickson 2005). The case of third powers corresponds to numbers having exactly six divisors, the case of forth powers to numbers having eight divisors, and so on.

 forms Sloane 2 , A007422 6, 8, 10, 14, 15, 21, 22, ... 3 , A030515 12, 18, 20, 28, 32, 44, ... 4 , , A030626 24, 30, 40, 42, 54, 56, ... 5 , A030628 48, 80, 112, 162, 176, ...

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## References

Dickson, L. E. History of the Theory of Numbers, Vol. 1: Divisibility and Primality. New York: Dover, p. 58, 2005.Halcke, P. Exs. 150-152 in Deliciae Mathematicae; oder, Mathematisches sinnen-confect. Hamburg, Germany: N. Sauer, p. 197, 1719.Ireland, K. and Rosen, M. A Classical Introduction to Modern Number Theory, 2nd ed. New York: Springer-Verlag, p. 19, 1990.Kaplansky, I. "The First Two Chapters of Dickson's History." Unpublished manuscript, Apr. 1999.Lionnet, E. "Note sur les nombres parfaits." Nouv. Ann. Math. 18, 306-308, 1879.Lucas, E. Ex. 6 in Théorie des nombres. Paris: Gauthier-Villars, p. 373, 1891.Sloane, N. J. A. Sequences A000040/M0652, A007422/M4068, A007955, A030515, A030626, A030628, A048943, A048944, A048945, and A048946 in "The On-Line Encyclopedia of Integer Sequences."Smarandache, F. Only Problems, Not Solutions!, 4th ed. Phoenix, AZ: Xiquan, 1993.

Divisor Product

## Cite this as:

Weisstein, Eric W. "Divisor Product." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/DivisorProduct.html