Multiply perfect number
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.
It can be proven that:
The following table gives an overview of the smallest k-perfect numbers for k ≤ 11 (sequence in the OEIS):
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.
A number n with σ(n) = 2n is perfect.
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.
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