e-Perfect Number

A number  is called an -perfect number if , where  is the sum of the e-Divisors of . If  is squarefree, then . As a result, if  is -perfect and  is squarefree with , then  is -perfect.

The first few -perfect numbers are 36, 180, 252, 396, 468, ... (OEIS A054979). There are no odd -perfect numbers. The first few primitive -perfect numbers are 36, 1800, 2700, 17424, ... (OEIS A054980).

REFERENCES:

Guy, R. K. "Exponential-Perfect Numbers." §B17 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 73, 1994.

Sloane, N. J. A. Sequences A054979 and A054980 in "The On-Line Encyclopedia of Integer Sequences."

Subbarao, M. V. and Suryanarayan, D. "Exponential Perfect and Unitary Perfect Numbers." Not. Amer. Math. Soc. 18, 798, 1971.