# abundant number

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## English

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### Noun

abundant number (plural abundant numbers)

1. (number theory) A number that is less than the sum of its proper divisors (all divisors except the number itself).
The factors of 30 are 1, 2, 3, 5, 6, 10, 15 and 30, and 1 + 2 + 3 + 5 + 6 + 10 + 15 = 42, which is greater than 30, so 30 is an abundant number.
• 1970, Geometric Transformations III, Random House, page 128,
It has been shown that the largest odd number which cannot be written as the sum of two abundant numbers is 20161.
• 1992, Stanley Rabinowitz (editor), Index to Mathematical Problems, 1980-1984, MathPro Press, page 185,
(a) Let k be fixed. Do there exist sequences of k consecutive abundant numbers?
• 1996, Richard R. Hall, Sets of Multiples, Cambridge University Press}, page xi,
We shall not be concerned with abundant numbers in this book, nevertheless it may be helpful to use this historical example as an illustration. We note the property that any multiple of an abundant number is abundant.

#### Usage notes

• The requirement may be expressed as ${\displaystyle s(n)>n}$, where ${\displaystyle s(n)}$ denotes the aliquot sum (sum of proper divisors) of ${\displaystyle n}$.
• It is also sometimes expressed as ${\displaystyle \sigma (n)>2n}$, where ${\displaystyle \sigma (n)}$ (sometimes ${\displaystyle \sigma _{1}(n)}$) denotes the sum of all divisors of ${\displaystyle n}$.
• Given an abundant number ${\displaystyle n}$, the amount, ${\displaystyle s(n)-n,}$ by which the aliquot sum exceeds it may be called its abundance.
• For arbitrary ${\displaystyle n}$, the ratio ${\displaystyle {\frac {\sigma (n)}{n}}}$ may be called its abundancy index. Thus, an abundant number is one whose abundancy index is > 2.

#### Synonyms

##### Hyponyms
• (number that is less than the sum of its proper divisors): weird number