Multiply perfect number

Demonstration, with Cuisenaire rods, of the 2-perfection of the number 6

In mathematics, a multiply perfect number (also called multiperfect number or pluperfect number) is a generalization of a perfect number.

For a given natural number k, a number n is called k-perfect (or k-fold perfect) if and only if the sum of all positive divisors of n (the divisor function, σ(n)) is equal to kn; a number is thus perfect if and only if it is 2-perfect. A number that is k-perfect for a certain k is called a multiply perfect number. As of 2014, k-perfect numbers are known for each value of k up to 11.^[1]

It can be proven that:

• For a given prime number p, if n is p-perfect and p does not divide n, then pn is (p+1)-perfect. This implies that an integer n is a 3-perfect number divisible by 2 but not by 4, if and only if n/2 is an odd perfect number, of which none are known.
• If 3n is 4k-perfect and 3 does not divide n, then n is 3k-perfect.

Smallest k-perfect numbers

The following table gives an overview of the smallest k-perfect numbers for k ≤ 11 (sequence A007539 in OEIS):

k	Smallest k-perfect number	Found by
1	1	ancient
2	6	ancient
3	120	ancient
4	30240	René Descartes, circa 1638
5	14182439040	René Descartes, circa 1638
6	154345556085770649600	Robert Daniel Carmichael, 1907
7	141310897947438348259849402738485523264343544818565120000	TE Mason, 1911
8	2.34111439263306338... × 10^161	Paul Poulet, 1929[1]
9	7.9842491755534198... × 10^465	Fred Helenius[1]
10	2.86879876441793479... × 10^923	Ron Sorli[1]
11	2.51850413483992918... × 10^1906	George Woltman[1]

For example, 120 is 3-perfect because the sum of the divisors of 120 is
1+2+3+4+5+6+8+10+12+15+20+24+30+40+60+120 = 360 = 3 × 120.

Properties

• The number of multiperfect numbers less than X is $o(X^{\epsilon})$ for all positive ε.^[2]
• The only known odd multiply perfect number is 1.

Specific values of k

Perfect numbers

A number n with σ(n) = 2n is perfect.

Triperfect numbers

A number n with σ(n) = 3n is triperfect. An odd triperfect number must exceed 10⁷⁰, have at least 12 distinct prime factors, the largest exceeding 10⁵.^[3]

References

1. Flammenkamp
2. Sándor et al (2006) p.105
3. Sandor et al (2006) pp.108–109

• Flammenkamp, Achim. "The Multiply Perfect Numbers Page". Retrieved 22 January 2014.
• Laatsch, Richard (1986). "Measuring the abundancy of integers". Mathematics Magazine 59 (2): 84–92. ISSN 0025-570X. JSTOR 2690424. MR 0835144. Zbl 0601.10003.
• Kishore, Masao (1987). "Odd triperfect numbers are divisible by twelve distinct prime factors". J. Aust. Math. Soc. Ser. A 42 (2): 173–182. doi:10.1017/s1446788700028184. ISSN 0263-6115. Zbl 0612.10006.
• Merickel, James G. (1999). "Problem 10617 (Divisors of sums of divisors)". Am. Math. Monthly 106 (7): 693. JSTOR 2589515. MR 1543520.
• Weiner, Paul A. (2000). "The abundancy ratio, a measure of perfection". Math. Mag. 73 (4): 307–310. JSTOR 2690980. MR 1573474.
• Sorli, Ronald M. (2003), Algorithms in the study of multiperfect and odd perfect numbers
• Ryan, Richard F. (2003). "A simpler dense proof regarding the abundancy index". Math. Mag. 76 (4): 299–301. JSTOR 3219086. MR 1573698.
• Guy, Richard K. (2004). Unsolved problems in number theory (3rd ed.). Springer-Verlag. B2. ISBN 978-0-387-20860-2. Zbl 1058.11001.
• Broughan, Kevin A.; Zhou, Qizhi (2008). "Odd multiperfect numbers of abundancy 4". J. Number Theory 126 (6): 1566–1575. doi:10.1016/j.jnt.2007.02.001. MR 2419178.
• Sándor, József; Mitrinović, Dragoslav S.; Crstici, Borislav, eds. (2006). Handbook of number theory I. Dordrecht: Springer-Verlag. ISBN 1-4020-4215-9. Zbl 1151.11300.
• Sándor, Jozsef; Crstici, Borislav, eds. (2004). Handbook of number theory II. Dordrecht: Kluwer Academic. pp. 32–36. ISBN 1-4020-2546-7. Zbl 1079.11001.

External links

• The Multiply Perfect Numbers page
• The Prime Glossary: Multiply perfect numbers