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 (Sloane's A000396). The number 120 was long known to be 3-multiply perfect () since

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

2 | Sloane's A000396 | 6, 28, 496, 8128, ..., |

3 | Sloane's A005820 | 120, 672, 523776, 459818240, 1476304896, 51001180160 |

4 | Sloane's A027687 | 30240, 32760, 2178540, 23569920, ... |

5 | Sloane's A046060 | 14182439040, 31998395520, 518666803200, ... |

6 | Sloane's A046061 | 154345556085770649600, 9186050031556349952000, ... |

In 1900-1901, Lehmer proved that has at least three distinct Prime factors, has at least four, at least six, at least nine, and at least 14.

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An outdated database is maintained by R. Schroeppel, who lists 2,094 multiperfects, and an up-to-date list by J. L. Moxham (1998). 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, 1101, 1129, 46, 0, 0, ....

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.

**References**

Brown, A. L. ``Multiperfect Numbers.'' *Scripta Math.* **20**, 103-106, 1954.

Dickson, L. E. *History of the Theory of Numbers, Vol. 1: Divisibility and Primality.* New York: Chelsea, pp. 33-38, 1952.

Flammenkamp, A. ``Multiply Perfect Numbers.'' http://www.uni-bielefeld.de/~achim/mpn.html.

Franqui, B. and García, M. ``Some New Multiply Perfect Numbers.'' *Amer. Math. Monthly* **60**, 459-462, 1953.

Franqui, B. and García, M. ``57 New Multiply Perfect Numbers.'' *Scripta Math.* **20**, 169-171, 1954.

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.

Helenius, F. W. ``Multiperfect Numbers (MPFNs).'' http://pweb.netcom.com/~fredh/mpfn/.

Madachy, J. S. *Madachy's Mathematical Recreations.* New York: Dover, pp. 149-151, 1979.

Moxham, J. L. ``13 New MPFN's.'' math-fun@cs.arizona.edu posting, Aug 13, 1998.

Poulet, P. *La Chasse aux nombres,* Vol. 1. Brussels, pp. 9-27, 1929.

Schroeppel, R. ``Multiperfect Numbers-Multiply Perfect Numbers-Pluperfect Numbers-MPFNs.'' Rev. Dec. 13, 1995. ftp://ftp.cs.arizona.edu/xkernel/rcs/mpfn.html.

Schroeppel, R. (moderator). mpfn mailing list. e-mail `rcs@cs.arizona.edu` to subscribe.

Sloane, N. J. A. Sequences A027687, A046060, A046061, A000396/M4186, and A005820/M5376 in ``An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html.

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1999-05-26