Multiply perfect number
Encyclopedia
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 divisor
s 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 July 2004, k-perfect numbers are known for each value of k up to 11.
It can be proven that:
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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a multiply perfect number (also called multiperfect number or pluperfect number) is a generalization of a perfect number
Perfect number
In number theory, a perfect number is a positive integer that is equal to the sum of its proper positive divisors, that is, the sum of its positive divisors excluding the number itself . Equivalently, a perfect number is a number that is half the sum of all of its positive divisors i.e...
.
For a given natural number
Natural number
In mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...
k, a number n is called k-perfect (or k-fold perfect) if and only if
If and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
the sum of all positive divisor
Divisor
In mathematics, a divisor of an integer n, also called a factor of n, is an integer which divides n without leaving a remainder.-Explanation:...
s of n (the divisor function
Divisor function
In mathematics, and specifically in number theory, a divisor function is an arithmetical function related to the divisors of an integer. When referred to as the divisor function, it counts the number of divisors of an integer. It appears in a number of remarkable identities, including relationships...
, σ(n)) is equal to kn; a number is thus perfect
Perfect number
In number theory, a perfect number is a positive integer that is equal to the sum of its proper positive divisors, that is, the sum of its positive divisors excluding the number itself . Equivalently, a perfect number is a number that is half the sum of all of its positive divisors i.e...
if and only if
If and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
it is 2-perfect. A number that is k-perfect for a certain k is called a multiply perfect number. As of July 2004, k-perfect numbers are known for each value of k up to 11.
It can be proven that:
- For a given prime numberPrime numberA prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
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 numberPerfect numberIn number theory, a perfect number is a positive integer that is equal to the sum of its proper positive divisors, that is, the sum of its positive divisors excluding the number itself . Equivalently, a perfect number is a number that is half the sum of all of its positive divisors i.e...
, 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 <= 7 (cf. Sloane's A007539):| k | Smallest k-perfect number | Found by |
|---|---|---|
| 1 | 1 | ancient |
| 2 | 6 | ancient |
| 3 | 120 120 (number) 120 is the natural number following 119 and preceding 121. 120 was known as "the great hundred", especially prior to the year 1700, from the Teutonic Hundert which equalled 120. The number 100, now known commonly as "one hundred" was then known as "the small hundred". It is also known as... |
ancient |
| 4 | 30240 | René Descartes René Descartes René Descartes ; was a French philosopher and writer who spent most of his adult life in the Dutch Republic. He has been dubbed the 'Father of Modern Philosophy', and much subsequent Western philosophy is a response to his writings, which are studied closely to this day... , circa 1638 |
| 5 | 14182439040 | René Descartes, circa 1638 |
| 6 | 154345556085770649600 | Robert Daniel Carmichael Robert Daniel Carmichael Robert Daniel Carmichael was a leading American mathematician. Carmichael was born in Goodwater, Alabama. He attended Lineville College, briefly, and he earned his bachelor's degree in 1898, while he was studying towards his Ph.D. degree at Princeton University. Carmichael completed the... , 1907 |
| 7 | 141310897947438348259849402738 485523264343544818565120000 | TE Mason, 1911 |
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.