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Divisor Product


By analogy with the divisor function sigma_1(n), let

 pi(n)=product_(d|n)d
(1)

denote the product of the divisors d of n (including n itself). For n=1, 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

 pi(n)=n^(sigma_0(n)/2).
(2)

The following table gives values of n for which pi(n) is a Pth power. Lionnet (1879) considered the case P=2.

POEISn
2A0489431, 6, 8, 10, 14, 15, 16, 21, 22, 24, 26, ...
3A0489441, 4, 8, 9, 12, 18, 20, 25, 27, 28, 32, ...
4A0489451, 24, 30, 40, 42, 54, 56, 66, 70, 78, ...
5A0489461, 16, 32, 48, 80, 81, 112, 144, 162, ...

Write the prime factorization of a number n,

 n=p_1^(a_1)p_2^(a_2)...p_r^(a_r).
(3)

Then the power of p_i occurring in pi(n) is

 1/2a_i(a_1+1)(a_2+1)...(a_r+1)
(4)

(Kaplansky 1999). This allows rules for determining when pi(n) is a power of n to be determined, as considered by Halcke (1719) and Lionnet (1879). Let p, q, and r be distinct primes, then the following table gives the conditions and first few n for which pi(n) is a given power P of n (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.

PformsSloanen
2p^3, pqA0074226, 8, 10, 14, 15, 21, 22, ...
3p^5, p^2qA03051512, 18, 20, 28, 32, 44, ...
4p^7, p^3q, pqrA03062624, 30, 40, 42, 54, 56, ...
5p^9, p^4qA03062848, 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 Unpublished manuscript, Apr. 1999.Lionnet, E. "Note sur les nombres parfaits."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.

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Divisor Product

Cite this as:

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

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