A hyperperfect number for a given is an integer such that , where is the sum of divisors function. is then called -hyperperfect. For example, 325 is 3-hyperperfect since its divisors (1, 5, 13, 25, 65, 325) add up to 434, and . Numbers that are 1-hyperperfect are by default called perfect numbers, since .
The 2-hyperperfect numbers are listed in A007593 of Sloane’s OEIS. As of 2007, the only known 3-hyperperfect number is 325. The two known 4-hyperperfect numbers are 1950625 and 1220640625, a sequence too short to list in the OEIS, and no 5-hyperperfect numbers are known to exist. The 6-hyperperfect numbers are listed in A028499.
- 1 Judson S. McCrainie, “A Study of Hyperperfect Numbers” Journal of Integer Sequences 3 (2000): 00.1.3
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