The first few abundant numbers are 12, 18, 20, 24, 30, 36, ... (OEIS A005101).
Every positive integer
with
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
(2)
That 945 is abundant can be seen by computing
(3)
Define the density function
(4)
(correcting the expression in Finch 2003, p. 126) for a positivereal number where gives the cardinal number
of the set ,
then Davenport (1933) proved that exists and is continuous for all , and Erdős (1934) gave a simplified proof (Finch 2003).
The special case
then gives the asymptotic density of abundant numbers,
(5)
The following table summarizes improvements in bounds on the constant over time.
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for which
is the divisor function and 
is the restricted divisor function. The quantity 
is sometimes called the abundance.
with 
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.
where 
gives the cardinal number
of the set 
,
then Davenport (1933) proved that 
exists and is continuous for all 
, and Erdős (1934) gave a simplified proof (Finch 2003).
The special case 
then gives the asymptotic density of abundant numbers,
