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 the 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... × 10¹⁶¹	Paul Poulet, 1929^[1]
9	7.9842491755534198... × 10⁴⁶⁵	Fred Helenius^[1]
10	2.86879876441793479... × 10⁹²³	Ron Sorli^[1]
11	2.51850413483992918... × 10¹⁹⁰⁶	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

Main article: Perfect number

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 2 3 4 5 Flammenkamp
↑ Sándor et al (2006) p.105
↑ 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

Divisibility-based sets of integers

Overview	Integer factorization Divisor Unitary divisor Divisor function Prime factor Fundamental theorem of arithmetic Arithmetic number

Factorization forms	Prime Composite Semiprime Pronic Sphenic Square-free Powerful Perfect power Achilles Smooth Regular Rough Unusual

Constrained divisor sums	Perfect Almost perfect Quasiperfect Multiply perfect Hemiperfect Hyperperfect Superperfect Unitary perfect Semiperfect Practical Erdős–Nicolas

With many divisors	Abundant Primitive abundant Highly abundant Superabundant Colossally abundant Highly composite Superior highly composite Weird

Aliquot sequence-related	Untouchable Amicable Sociable Betrothed

Other sets	Deficient Friendly Solitary Sublime Harmonic divisor Frugal Equidigital Extravagant

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a × 2^b ± 1

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2-dimensional

centered	Centered triangular Centered square Centered pentagonal Centered hexagonal Centered heptagonal Centered octagonal Centered nonagonal Centered decagonal Star

non-centered	Triangular Square Square triangular Pentagonal Hexagonal Heptagonal Octagonal Nonagonal Decagonal Dodecagonal

3-dimensional

centered	Centered tetrahedral Centered cube Centered octahedral Centered dodecahedral Centered icosahedral

non-centered	Tetrahedral Octahedral Dodecahedral Icosahedral Stella octangula

pyramidal	Square pyramidal Pentagonal pyramidal Hexagonal pyramidal Heptagonal pyramidal

4-dimensional

centered	Centered pentachoric Squared triangular

non-centered	Pentatope

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By properties of σ(n)	Abundant Almost perfect Arithmetic Colossally abundant Descartes Hemiperfect Highly abundant Highly composite Hyperperfect Multiply perfect Perfect Practical number Primitive abundant Quasiperfect Refactorable Sublime Superabundant Superior highly composite Superperfect

By properties of Ω(n)	Almost prime Semiprime

By properties of φ(n)	Highly cototient Highly totient Noncototient Nontotient Perfect totient Sparsely totient

By properties of s(n)	Amicable Betrothed Deficient Semiperfect

Dividing a quotient

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