Hyperperfect Number

A number  is called -hyperperfect if

			(1)
			(2)

where  is the divisor function and the summation is over the proper divisors with . Rearranging gives

(3)

Taking  gives the usual perfect numbers.

If  is an odd integer, and  and  are prime, then  is -hyperperfect. McCranie (2000) conjectures that all -hyperperfect numbers for odd  are in fact of this form. Similarly, if  and  are distinct odd primes such that  for some integer , then  is -hyperperfect. Finally, if  and  is prime, then if  is prime for some < then  is -hyperperfect (McCranie 2000).

The first few hyperperfect numbers (excluding perfect numbers) are 21, 301, 325, 697, 1333, ... (OEIS A007592). If perfect numbers are included, the first few are 6, 21, 28, 301, 325, 496, ... (OEIS A034897), whose corresponding values of  are 1, 2, 1, 6, 3, 1, 12, ... (OEIS A034898). The following table gives the first few -hyperperfect numbers for small values of . McCranie (2000) has tabulated all hyperperfect numbers less than .

	OEIS	-hyperperfect number
1	A000396	6 ,28, 496, 8128, ...
2	A007593	21, 2133, 19521, 176661, ...
3		325, ...
4		1950625, 1220640625, ...
6	A028499	301, 16513, 60110701, ...
10		159841, ...
11		10693, ...
12	A028500	697, 2041, 1570153, 62722153, ...

REFERENCES:

Guy, R. K. "Almost Perfect, Quasi-Perfect, Pseudoperfect, Harmonic, Weird, Multiperfect and Hyperperfect Numbers." §B2 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 45-53, 1994.

McCranie, J. S. "A Study of Hyperperfect Numbers." J. Integer Sequences 3, No. 00.1.3, 2000. https://www.math.uwaterloo.ca/JIS/VOL3/VOL3/mccranie.

Minoli, D. "Issues in Nonlinear Hyperperfect Numbers." Math. Comput. 34, 639-645, 1980.

Roberts, J. The Lure of the Integers. Washington, DC: Math. Assoc. Amer., p. 177, 1992.

Sloane, N. J. A. Sequences A000396/M4186, A007592/M5113, A007593/M5121, A028499, A028500, A034897, and A034898 in "The On-Line Encyclopedia of Integer Sequences."

te Riele, H. J. J. "Hyperperfect Numbers with Three Different Prime Factors." Math. Comput. 36, 297-298, 1981.