- Multiply perfect number
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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 July 2004, k-perfect numbers are known for each value of k up to 11.
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 <= 7 (cf. Sloane's A007539):
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 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.External links
Divisibility-based sets of integers Overview Forms of factorization Constrained divisor sums Perfect number · Almost perfect number · Quasiperfect number · Multiply perfect number · Hyperperfect number · Superperfect number · Unitary perfect number · Semiperfect number · Primitive semiperfect number · Practical numberNumbers with many divisors Abundant number · Primitive abundant number · Highly abundant number · Superabundant number · Colossally abundant number · Highly composite number · Superior highly composite number · Weird numberNumbers related
to aliquot sequencesOther Categories:- Integer sequences
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