Abundant number

English

Pronunciation

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Noun

1. Template:Lb 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.

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   • 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?

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Usage notes

• The requirement may be expressed as $s(n)>n$, where $s(n)$ denotes the Template:M (sum of proper divisors) of $n$.
  • It is also sometimes expressed as $\sigma (n)>2n$, where $\sigma (n)$ (sometimes ${\sigma }_1(n)$) denotes the sum of all divisors of $n$.
• Given an abundant number $n$, the amount, $s(n)-n,$ by which the aliquot sum exceeds it may be called its Template:M.
• For arbitrary $n$, the ratio $\frac{\sigma (n)}{n}$ may be called its Template:M. Thus, an abundant number is one whose abundancy index is > 2.

Synonyms

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Hyponyms

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Derived terms

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Translations

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See also

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Further reading

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• abundant number on The Prime Glossary
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