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,
