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# Even abundant numbers

The even abundant numbers are even numbers ${\displaystyle \scriptstyle n\,}$ whose sum of divisors exceeds ${\displaystyle \scriptstyle 2n\,}$. (Or whose sum of proper divisors exceeds ${\displaystyle \scriptstyle n\,}$.)

A173490 Even abundant numbers.

{12, 18, 20, 24, 30, 36, 40, 42, 48, 54, 56, 60, 66, 70, 72, 78, 80, 84, 88, 90, 96, 100, 102, 104, 108, 112, 114, 120, 126, 132, 138, 140, 144, 150, 156, 160, 162, 168, 174, 176, 180, 186, 192, 196, 198, ...}

The first 231 terms are the same as in A005101 (abundant numbers).

While the first even abundant number is ${\displaystyle \scriptstyle 12\,=\,2^{2}\cdot 3\,}$ with ${\displaystyle \scriptstyle \sigma (12)\,=\,{\frac {2^{3}-1}{2-1}}\cdot (3+1)\,=\,7\cdot 4\,=\,28\,>\,24\,=\,2\cdot 12\,}$, the first odd abundant number is ${\displaystyle \scriptstyle 945\,=\,3^{3}\cdot 5\cdot 7\,}$ with ${\displaystyle \scriptstyle \sigma (945)\,=\,{\frac {3^{4}-1}{3-1}}\cdot (5+1)\cdot (7+1)\,=\,40\cdot 6\cdot 8\,=\,1920\,>\,1890\,=\,2\cdot 945\,}$!

The first odd abundant number is the 232 nd abundant number!

Some abundant numbers (and even abundant numbers):

• Every positive multiple greater than 1 of a perfect number is abundant (those are all even, unless odd perfect numbers exist...).
• Every positive multiple of an abundant number is abundant.
• Every positive multiple of an even abundant number is even and abundant.
• Every even positive multiple of an odd abundant number is even and abundant.

## Properties

Abundant numbers (and hence even abundant numbers) are closed under multiplication by arbitrary positive integers: any positive multiple of an abundant number is abundant.

## Asymptotic density

As a consequence of the closure under multiplication by arbitrary positive integers, the even abundant numbers are of positive density. In particular, their lower density is at least 0.2453 and their upper density is at most 0.2460.[1][2]

• A005231 Odd abundant numbers (odd numbers whose sum of divisors of ${\displaystyle \scriptstyle n\,}$ exceeds ${\displaystyle \scriptstyle 2n\,}$).