Numbers and numerals with interesting properties

Math › Numbers › Number Theory › Kinds of Numbers

Perfect number

https://en.wikipedia.org/wiki/Perfect_number

In number theory, a perfect number is a positive integer that is equal to its aliquot sum. An aliquot sum of an integer is the sum of its positive factors, excluding the integer itself.

a perfect number is a positive integer, $n$, such that the sum of its positive divisors is $2n$.

Officially, the number is equal to the sum of its positive divisors, excluding the number itself (although it seems dodgy to accept 1, but exclude the number itself from the list of its factors).

Number $n$ is perfect if the sum of its factors is $2n$.

The sum of divisors of a number, excluding the number itself, is called its aliquot sum, so

https://en.wikipedia.org/wiki/Aliquot_sum

Equivalently, a perfect number is a number that is half the sum of all of its positive divisors including itself i.e. $\sigma_1(n) = 2n$

For instance, 28 is perfect as 1 + 2 + 4 + 7 + 14 + 28 = 56 = 2 × 28.

Let $f$ be a function that produces a set of factors of an integer: $f(n) = \{m_1, m_2, \dots, m_{k-1}, m_k\}$ The first factor, $m_1$, is always 1, and the last factor, $m_k$, is always the integer itself, $n$.

$\displaystyle f(n) = \sum_{i=1}^k m_i = 2n$

$\displaystyle \sum f(n) = 2n$

Examples:

6 = 1+2+3

28= 1.2.4.7.14

496=1+2+4+8+16+21+62+124+248

8128= 1+2+4+8+16+32+64+127+256+308+1016+2032+4064

There's 1 perfect number for every j-digits

In 2011: 47 discovered

The conjecture about infinity ofPN is unknown.