abundant number

Noun

 * 1)  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.
 * 1) * 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?
 * (a) Let k be fixed. Do there exist sequences of k consecutive abundant numbers?

Usage notes

 * The requirement may be expressed as $$s(n)>n$$, where $$s(n)$$ denotes the (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.
 * For arbitrary $$n$$, the ratio $$\frac{\sigma(n)}{n}$$ may be called its . Thus, an abundant number is one whose abundancy index is > 2.

Translations

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 * Italian:
 * Spanish: número abundante
 * Swedish: ,