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List of Mersenne primes and perfect numbers

Mersenne primes and perfect numbers are two deeply interlinked types of natural numbers in number theory. Mersenne primes, named after the friar Marin Mersenne, are prime numbers that can be expressed

Perfect number

In number theory, a perfect number is a positive integer that is equal to the sum of its positive divisors, excluding the number itself. For instance, 6 has divisors 1, 2 and 3 (excluding itself), and

Unitary perfect number

A unitary perfect number is an integer which is the sum of its positive proper unitary divisors, not including the number itself (a divisor d of a number n is a unitary divisor if d and n/d share no c

Hyperperfect number

In mathematics, a k-hyperperfect number is a natural number n for which the equality n = 1 + k(σ(n) − n − 1) holds, where σ(n) is the divisor function (i.e., the sum of all positive divisors of n). A

Harmonic divisor number

In mathematics, a harmonic divisor number, or Ore number (named after Øystein Ore who defined it in 1948), is a positive integer whose divisors have a harmonic mean that is an integer. The first few h

Semiperfect number

In number theory, a semiperfect number or pseudoperfect number is a natural number n that is equal to the sum of all or some of its proper divisors. A semiperfect number that is equal to the sum of al

Aliquot sum

In number theory, the aliquot sum s(n) of a positive integer n is the sum of all proper divisors of n, that is, all divisors of n other than n itself.That is, It can be used to characterize the prime

Mersenne prime

In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form Mn = 2n − 1 for some integer n. They are named after Marin Mersenne,

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 (

Hemiperfect number

In number theory, a hemiperfect number is a positive integer with a half-integer abundancy index. In other words, σ(n)/n = k/2 for an odd integer k, where σ(n) is the divisor function, the sum of all

Euclid–Euler theorem

The Euclid–Euler theorem is a theorem in number theory that relates perfect numbers to Mersenne primes. It states that an even number is perfect if and only if it has the form 2p−1(2p − 1), where 2p −

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