Multiperfect Number

A number is -multiperfect (also called a -multiply perfect number or -pluperfect number) if

for some integer , where is the divisor function. The value of is called the class. The special case corresponds to perfect numbers , which are intimately connected with Mersenne primes (OEIS A000396). The number 120 was long known to be 3-multiply perfect () since

The following table gives the first few for , 3, ..., 6.

Lehmer (1900-1901) proved that has at least three distinct prime factors, has at least four, at least six, at least nine, and at least 14, etc.

As of 1911, 251 pluperfect numbers were known (Carmichael and Mason 1911). As of 1929, 334 pluperfect numbers were known, many of them found by Poulet. Franqui and García (1953) found 63 additional ones (five s, 29 s, and 29 s), several of which were known to Poulet but had not been published, bringing the total to 397. Brown (1954) discovered 110 pluperfects, including 31 discovered but not published by Poulet and 25 previously published by Franqui and García (1953), for a total of 482. Franqui and García (1954) subsequently discovered 57 additional pluperfects (3 s, 52 s, and 2 s), increasing the total known to 539.

An outdated database is maintained by R. Schroeppel, who lists multiperfects, and up-to-date lists by J. L. Moxham and A. Flammenkamp. It is believed that all multiperfect numbers of index 3, 4, 5, 6, and 7 are known. The number of known -multiperfect numbers are 1, 37, 6, 36, 65, 245, 516, 1134, 2036, 644, 1, 0, ... (Moxham 2001, Flammenkamp, Woltman 2000). Moxham (2000) found the largest known multiperfect number, approximately equal to , on Feb. 13, 2000.

If is a number such that , then is a number. If is a number such that , then is a number. If is a number such that 3 (but not 5 and 9) divides , then is a number.

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