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, ...

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 Unpublished manuscript, Apr. 1999.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.