Quasiperfect number

Template:Short description In mathematics, a quasiperfect number is a natural number Template:Mvar for which the sum of all its divisors (the sum-of-divisors function $\sigma (n)$) is equal to $2n+1$. 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.

Related

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

Numbers Template:Mvar whose sum of factors equals $2n+2$ are known to exist. They are of form $2^{n-1}\times (2^n-3)$ where $2^n-3$ is a prime. The only exception known so far is $650=2\times 5^2\times 13$. They are 20, 104, 464, 650, 1952, 130304, 522752, ... (sequence A088831 in the OEIS). Numbers Template:Mvar whose sum of factors equals $2n-2$ are also known to exist. They are of form $2^{n-1}\times (2^n+1)$ where $2^n+1$ 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 (sequence A191363 in the OEIS)

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

References

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