Multiperfect Number

A number $n$ is $k$-multiperfect (also called a $k$-Multiply Perfect Number or $k$-Pluperfect Number) if

$$\sigma(n)=kn$$

for some Integer $k>2$, where $\sigma(n)$ is the Divisor Function. The value of $k$ is called the Class. The special case $k=2$ corresponds to Perfect Numbers $P_2$, which are intimately connected with Mersenne Primes (Sloane's A000396). The number 120 was long known to be 3-multiply perfect ($P_3$) since

$$\sigma(120)=3\cdot 120.$$

The following table gives the first few $P_n$ for $n=2$, 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 $P_3$ has at least three distinct Prime factors, $P_4$ has at least four, $P_5$ at least six, $P_6$ at least nine, and $P_7$ 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 $P_5$s, 29 $P_6$s, and 29 $P_7$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 $P_6$s, 52 $P_7$s, and 2 $P_8$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An outdated database is maintained by R. Schroeppel, who lists 2,094 multiperfects, and an up-to-date list by J. L. Moxham (1998).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 $n$-multiperfect numbers are 1, 37, 6, 36, 65, 245, 516, 1101, 1129, 46, 0, 0, ....

If $n$ is a $P_5$ number such that $3\notdiv n$, then $3n$ is a $P_4$ number. If $3n$ is a $P_{4k}$ number such that $3\notdiv n$, then $n$ is a $P_{3k}$ number. If $n$ is a $P_3$ number such that 3 (but not 5 and 9) Divides $n$, then $45n$ is a $P_4$ number.

See also e-Multiperfect Number, Friendly Pair, Hyperperfect Number, Infinary Multiperfect Number, Mersenne Prime, Perfect Number, Unitary Multiperfect 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.