8128 is the

integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...

following 8127 and preceding 8129. It is most notable for being a

perfect number 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 perfec ...

(its proper divisors 1, 2, 4, 8, 16, 32, 64, 127, 254, 508, 1016, 2032, and 4064 add up to 8128), and one of the earliest numbers to be recognized as such. As a perfect number, it is tied to the

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 for some integer . They are named after Marin Mersenne, a French Minim friar, who studied them in the early 1 ...

127, 2⁷ – 1, with 2⁶ (2⁷ – 1) yielding 8128. Also related to its being a perfect number, 8128 is a

harmonic divisor number In mathematics, a harmonic divisor number or Ore number is a positive integer whose divisors have a harmonic mean that is an integer. The first few harmonic divisor numbers are : 1, 6, 28, 140, 270, 496, 672, 1638, 2970, 6200, 8128, 8190 . ...

. Another consequence of 8128 being a perfect number is that it has the same prime factors as the sum of its divisors, its

cototient In number theory, Euler's totient function counts the positive integers up to a given integer that are relatively prime to . It is written using the Greek letter phi as \varphi(n) or \phi(n), and may also be called Euler's phi function. In othe ...

is a power of two, and it is a harmonic seed number (though there are deficient and abundant numbers that share these properties). 8128 is the 127th

triangular number A triangular number or triangle number counts objects arranged in an equilateral triangle. Triangular numbers are a type of figurate number, other examples being square numbers and cube numbers. The th triangular number is the number of dots in ...

, the 64th

hexagonal number A hexagonal number is a figurate number. The ''n''th hexagonal number ''h'n'' is the number of ''distinct'' dots in a pattern of dots consisting of the ''outlines'' of regular hexagons with sides up to n dots, when the hexagons are overlaid so ...

, a

happy number In number theory, a happy number is a number which eventually reaches 1 when the number is replaced by the sum of the square of each digit. For instance, 13 is a happy number because 1^2+3^2=10, and 1^2+0^2=1. On the other hand, 4 is not a happy ...

, the eighth 292

-gonal number In mathematics, a polygonal number is a number that counts dots arranged in the shape of a regular polygon. These are one type of 2-dimensional figurate numbers. Polygonal numbers were first studied during the 6th century BC by the Ancient Greeks, ...

, and the fourth 1356-gonal number, as well as the 43rd

centered nonagonal number A centered nonagonal number, (or centered enneagonal number), is a centered figurate number that represents a nonagon with a dot in the center and all other dots surrounding the center dot in successive nonagonal layers. The centered nonagonal ...

References

Integers {{Num-stub

See also

References