An abundant number, sometimes also called an excessive number, is a positive integer n for which
s(n)=sigma(n)-n>n,
(1)
where sigma(n) is the divisor function and s(n) is the restricted divisor function. The quantity sigma(n)-2n is sometimes called the abundance.
A number which is abundant but for which all its proper divisors are deficient is called a primitive abundant number (Guy 1994, p. 46).
The first few abundant numbers are 12, 18, 20, 24, 30, 36, ... (OEIS A005101).
Every positive integer n with (mod n)60 is abundant. Any multiple of a perfect number or an abundant number is also abundant. Prime numbers are not abundant. Every number greater than 20161 can be expressed as a sum of two abundant numbers.
There are only 21 abundant numbers less than 100, and they are all even. The first odd abundant number is
945=3^3·7·5.
(2)
That 945 is abundant can be seen by computing
s(945)=975>945.
(3)
AbundantNumberDensity
Define the density function
A(x)=lim_(n->infty)(|{k<=n:sigma(k)>=xk}|)/n
(4)
(correcting the expression in Finch 2003, p. 126) for a positive real number x where |B| gives the cardinal number of the set B, then Davenport (1933) proved that A(x) exists and is continuous for all x, and Erdős (1934) gave a simplified proof (Finch 2003). The special case A(2) then gives the asymptotic density of abundant numbers,