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, ...

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