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 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 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.


  • The number of multiperfect numbers less than X is o(X^{\epsilon}) for all positive ε.[2]
  • The only known odd multiply perfect number is 1.[citation needed]

Specific values of k

Perfect numbers

Main article: Perfect number

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]


  1. 1.0 1.1 1.2 1.3 1.4 Flammenkamp
  2. Sándor et al (2006) p.105
  3. Sandor et al (2006) pp.108–109

External links