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.
| 2 |
A000396 |
6, 28, 496, 8128, ... |
| 3 |
A005820 |
120, 672, 523776, 459818240, 1476304896, 51001180160 |
| 4 |
A027687 |
30240, 32760, 2178540, 23569920, ... |
| 5 |
A046060 |
14182439040, 31998395520, 518666803200, ... |
| 6 |
A046061 |
154345556085770649600, 9186050031556349952000, ... |
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An outdated database is maintained by R. Schroeppel, who lists 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.
REFERENCES:
Beck, W. and Najar, R. "A Lower Bound for Odd Triperfects." Math. Comput. 38, 249-251, 1982.
Brown, A. L. "Multiperfect Numbers." Scripta Math. 20, 103-106, 1954.
Carmichael and Mason, T. E. Proc. Indian Acad. Sci., 257-270, 1911.
Cohen, G. L. and Hagis, P. Jr. "Results Concerning Odd Multiperfect Numbers." Bull. Malaysian Math. Soc. 8, 23-26, 1985.
Dickson, L. E. History of the Theory of Numbers, Vol. 1: Divisibility and Primality. New York: Dover, pp. 33-38, 2005.
Flammenkamp, A. "Multiply Perfect Numbers." https://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)." https://home.netcom.com/~fredh/mpfn/.
Lehmer, D. N. Ann. Math. 2, 103-104, 1900-1901.
Madachy, J. S. Madachy's Mathematical Recreations. New York: Dover, pp. 149-151, 1979.
Moxham, J. L. "New Largest MPFN." mpfn@cs.arizona.edu posting, 13 Feb 2000.
Moxham, J. L. "1 New mpfns Total=4683." mpfn@cs.arizona.edu posting, 26 Mar 2001.
Perrier, J.-Y. "The Multi-Perfect Numbers." https://diwww.epfl.ch/~perrier/Multiparfaits.html
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 A000396/M4186, A005820/M5376, A027687, A046060, and A046061 in "The On-Line Encyclopedia of Integer Sequences."
Sorli, R. "Multiperfect Numbers." https://www-staff.maths.uts.edu.au/~rons/mpfn/mpfn.htm.
Woltman, G. "5 new MPFNs." mpfn@cs.arizona.edu posting, 23 Sep 2000.
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