A perfect number (integer) is equal to the sum of its divisors, including 1, and excluding itself. This has been around since Euclid. Recently, I noticed that at least for the initial integers, it is more common for that sum of divisors to be smaller than the number in question. However, for example, using the same rules the sum of the divisors of 12 is actually sixteen: The sum is here greater than the whole. Clearly, not perfect. Therefore, imperfect. But, has anyone studied these numbers?

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    $\begingroup$ see this Wikipedia article about abundant numbers $\endgroup$ Jan 3, 2020 at 0:23
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    $\begingroup$ The adjectives "deficient" and "abundant" (respectively) are used for such non-perfect numbers. $\endgroup$
    – hardmath
    Jan 3, 2020 at 0:24
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    $\begingroup$ I'm pretty sure all multiples of 6 greater than 6 are abundant because for any 6n, 3n, 2n, and n are all factors and they already sum up to 6n, so any extra factor such as 1 would make the sum exceed 6n. $\endgroup$ Jan 3, 2020 at 0:32
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    $\begingroup$ Yes, every multiple (beyond $1$) of a perfect number is abundant $\endgroup$ Jan 3, 2020 at 0:34
  • $\begingroup$ You might be interested in pairs of amicable numbers, each of which is the sum of proper divisors of the other. Eg. 220 and 284. Naturally one of these (the larger of the pair) is deficient and the other is abundant. $\endgroup$
    – hardmath
    Jan 3, 2020 at 0:39

1 Answer 1


The sum of all proper divisors of a number is less than, equal to, or greater than the number,

according as the number is deficient, perfect, or abundant.

The first 28 abundant numbers are $12, 18, 20, 24, 30, 36, 40, 42, 48, 54, 56, 60,$

$66, 70, 72, 78, 80, 84, 88, 90, 96, 100, 102, 104,$ $ 108, 112, 114, $ and $120.$

They are sequence A005101 in the On-Line Encyclopedia of Integer Sequences.

The smallest odd abundant number is $945$.

Every multiple (beyond $1$) of a perfect number is abundant.

You can read more about these numbers on Wikipedia.

  • $\begingroup$ In addition, every multiple of an abundant number is also abundant. $\endgroup$
    – gandalf61
    Jan 3, 2020 at 9:23

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