Trisha Shetty (Editor)

Multiply perfect number

Updated on
Edit
Like
Comment
Share on FacebookTweet on TwitterShare on LinkedInShare on Reddit
Multiply perfect number

In mathematics, a multiply perfect number (also called multiperfect number or pluperfect number) is a generalization of a perfect number.

Contents

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:

  • 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):

    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 o ( X ϵ ) for all positive ε.
  • The only known odd multiply perfect number is 1.

    • 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.

    References

    Multiply perfect number Wikipedia
    (Text) CC BY-SA