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In number theory, a perfect number is a positive integer that is equal to the sum of its positive proper divisors, that is, divisors excluding the number itself. For instance, 6 has proper divisors 1, 2 and 3, and 1 + 2 + 3 = 6, so 6 is a perfect number. The next perfect number is 28, since 1 + 2 + 4 + 7 + 14 = 28.
Perfect numbers are natural numbers that equal the sum of their positive proper divisors, which are divisors excluding the number itself. So, 6 is a perfect number because the proper divisors of 6 are 1, 2, and 3, and 1 + 2 + 3 = 6. [2] [4]
A perfect number is a natural number that equals the sum of its proper divisors, the numbers that are less than it and divide it evenly (with remainder zero). For instance, the proper divisors of 6 are 1, 2, and 3, which sum to 6, so 6 is perfect. A Mersenne prime is a prime number of the form M p = 2 p − 1, one less than a power of two.
The sum of positive divisors ... p 2 is 3); noting that 24 is the product of 2 3 ×3 1, a 1 is 3 and a 2 ... This function is used to recognize perfect numbers, ...
For a given natural number k, a number n is called k-perfect (or k-fold perfect) 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 ...
The sum of the reciprocals of all perfect powers including duplicates (but not including 1) equals 1. The Erdős–Moser equation, + + + = (+) where m and k are positive integers, is conjectured to have no solutions other than 1 1 + 2 1 = 3 1.
A perfect totient number is an integer that is equal to the sum of its iterated totients. That is, we apply the totient function to a number n, apply it again to the resulting totient, and so on, until the number 1 is reached, and add together the resulting sequence of numbers; if the sum equals n, then n is a perfect totient number.
The aliquot sum function can be used to characterize several notable classes of numbers: 1 is the only number whose aliquot sum is 0. A number is prime if and only if its aliquot sum is 1. [1] The aliquot sums of perfect, deficient, and abundant numbers are equal to, less than, and greater than the number itself respectively. [1]