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.

Example[]

The divisors of 120 are 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, and 120. Their sum is 360, which equals ${\displaystyle 3\times 120}$, so 120 is 3-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):

Properties[]

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

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 and have at least 12 distinct prime factors, the largest exceeding 10⁵.^[3]

Variations[]

Unitary multiply perfect numbers[]

A positive integer n is called a unitary multi k-perfect number if σ^*(n) = kn. A unitary multiply perfect number is simply a unitary multi k-perfect number for some positive integer k. Equivalently, unitary multiply perfect numbers are those n for which n divides σ^*(n). A unitary multi 2-perfect number is naturally called a unitary perfect number. In the case k > 2, no example of a unitary multi k-perfect number is known till now. It is known that if such a number exists, it must be even and greater than 10¹⁰² and must have more than forty four odd prime factors. This problem is probably very difficult to settle.

A divisor d of a positive integer n is called a unitary divisor if gcd(d, n/d) = 1. The concept of unitary divisor was originally due to R. Vaidyanathaswamy (1931) who called such a divisor as block factor. The present terminology is due to E. Cohen (1960). The sum of the (positive) unitary divisors of n is denoted by σ^*(n).

Bi-unitary multiply perfect numbers[]

A positive integer n is called a bi-unitary multi k-perfect number if σ^**(n) = kn. This concept is due to Peter Hagis (1987). A bi-unitary multiply perfect number is simply a bi-unitary multi k-perfect number for some positive integer k. Equivalently, bi-unitary multiply perfect numbers are those n for which n divides σ^**(n). A bi-unitary multi 2-perfect number is naturally called a bi-unitary perfect number, and a bi-unitary multi 3-perfect number is called a bi-unitary triperfect number.

A divisor d of a positive integer n is called a bi-unitary divisor of n if the greatest common unitary divisor (gcud) of d and n/d equals 1. This concept is due to D. Surynarayana (1972). The sum of the (positive) bi-unitary divisors of n is denoted by σ^**(n).

Peter Hagis (1987) proved that there are no odd bi-unitary multiperfect numbers. Haukkanen and Sitaramaiah (2020) found certain even bi-unitary triperfect numbers.

References[]

Sources[]

• Broughan, Kevin A.; Zhou, Qizhi (2008). "Odd multiperfect numbers of abundancy 4" (PDF). J. Number Theory. 126 (6): 1566–1575. doi:10.1016/j.jnt.2007.02.001. hdl:10289/1796. MR 2419178.
• Guy, Richard K. (2004). Unsolved problems in number theory (3rd ed.). Springer-Verlag. B2. ISBN 978-0-387-20860-2. Zbl 1058.11001.
• Haukkanen, Pentti; Sitaramaiah, V. (2020). "Bi-unitary multiperfect numbers, I" (PDF). Notes Number Theory Discrete Math. 26 (1): 93–171. doi:10.7546/nntdm.2020.26.1.93-171.
• Haukkanen, Pentti; Sitaramaiah, V. (2020). "Bi-unitary multiperfect numbers, II" (PDF). Notes Number Theory Discrete Math. 26 (2): 1–26. doi:10.7546/nntdm.2020.26.2.1-26.
• Haukkanen, Pentti; Sitaramaiah, V. (2020). "Bi-unitary multiperfect numbers, III" (PDF). Notes Number Theory Discrete Math. 26 (3): 33–67. doi:10.7546/nntdm.2020.26.3.33-67.
• Haukkanen, Pentti; Sitaramaiah, V. (2020). "Bi-unitary multiperfect numbers, IV(a)". Notes Number Theory Discrete Math. 26 (4): 2–32. doi:10.7546/nntdm.2020.26.4.2-32.
• 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.
• Laatsch, Richard (1986). "Measuring the abundancy of integers". Mathematics Magazine. 59 (2): 84–92. doi:10.2307/2690424. ISSN 0025-570X. JSTOR 2690424. MR 0835144. Zbl 0601.10003.
• Merickel, James G. (1999). "Problem 10617 (Divisors of sums of divisors)". Amer. Math. Monthly. 106 (7): 693. doi:10.2307/2589515. JSTOR 2589515. MR 1543520.
• Ryan, Richard F. (2003). "A simpler dense proof regarding the abundancy index". Math. Mag. 76 (4): 299–301. JSTOR 3219086. MR 1573698.
• 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.
• 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.
• Sorli, Ronald M. (2003). Algorithms in the study of multiperfect and odd perfect numbers (PhD thesis). Sydney: University of Technology. hdl:10453/20034.
• Weiner, Paul A. (2000). "The abundancy ratio, a measure of perfection". Math. Mag. 73 (4): 307–310. doi:10.1080/0025570x.2000.11996860. JSTOR 2690980. MR 1573474.

External links[]

• The Multiply Perfect Numbers page
• The Prime Glossary: Multiply perfect numbers
• Grime, James. "The Six Triperfect Numbers" (video). youTube. Brady Haran. Retrieved 29 June 2018.