Quasiperfect number

Template:Short description Template:Unsolved In mathematics, a quasiperfect number is a natural number Template:Mvar for which the sum of all its divisors (the sum-of-divisors function <math>\sigma(n)</math>) is equal to <math>2n + 1</math>. Equivalently, Template:Mvar is the sum of its non-trivial divisors (that is, its divisors excluding 1 and Template:Mvar). No quasiperfect numbers have been found so far.

The quasiperfect numbers are the abundant numbers of minimal abundance (which is 1).

Theorems

If a quasiperfect number exists, it must be an odd square number greater than 10³⁵ and have at least seven distinct prime factors.<ref>Template:Cite journal</ref>

Related

For a perfect number Template:Mvar the sum of all its divisors is equal to <math>2n</math>. For an almost perfect number Template:Mvar the sum of all its divisors is equal to <math>2n - 1</math>.

Numbers Template:Mvar whose sum of factors equals <math>2n + 2</math> are known to exist. They are of form <math>2^{n - 1} \times (2^n - 3)</math> where <math>2^n - 3</math> is a prime. The only exception known so far is <math>650 = 2 \times 5^2 \times 13</math>. They are 20, 104, 464, 650, 1952, 130304, 522752, ... Template:OEIS. Numbers Template:Mvar whose sum of factors equals <math>2n - 2</math> are also known to exist. They are of form <math>2^{n - 1} \times (2^n + 1)</math> where <math>2^n + 1</math> is prime. No exceptions are found so far. Because of the five known Fermat primes, there are five such numbers known: 3, 10, 136, 32896 and 2147516416 Template:OEIS

Betrothed numbers relate to quasiperfect numbers like amicable numbers relate to perfect numbers.

Notes

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References

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