Multiply perfect number
![](http://upload.wikimedia.org/wikipedia/commons/thumb/c/ca/Multiply_perfect_number_Cuisenaire_rods_6.png/220px-Multiply_perfect_number_Cuisenaire_rods_6.png)
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 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, have at least 12 distinct prime factors, the largest exceeding 105.[3]
References
- 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.