Difference between revisions of "Abundant number"
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An '''abundant number''' is a number <math>n</math> for which the sum of its [[proper divisor|proper divisors]] is greater than <math>n</math> itself. For example, 12 is abundant because the sum of its proper divisors is 1 + 2 + 3 + 4 + 6 = 16 > 12. | An '''abundant number''' is a number <math>n</math> for which the sum of its [[proper divisor|proper divisors]] is greater than <math>n</math> itself. For example, 12 is abundant because the sum of its proper divisors is 1 + 2 + 3 + 4 + 6 = 16 > 12. | ||
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==Examples== | ==Examples== | ||
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The first 10 abundant numbers are: | The first 10 abundant numbers are: | ||
12, 18, 20, 24, 30, 36, 40, 42, 48, 54,... | 12, 18, 20, 24, 30, 36, 40, 42, 48, 54,... | ||
| − | The first 10 odd abundant numbers are: | + | The first 10 [[odd integer|odd]] abundant numbers are: |
945, 1575, 2205, 2835, 3465, 4095, 4725, 5355, 5775, 5985,... | 945, 1575, 2205, 2835, 3465, 4095, 4725, 5355, 5775, 5985,... | ||
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There are infinitely many abundant numbers. | There are infinitely many abundant numbers. | ||
| − | Every positive multiple of an abundant number is an abundant number. | + | Every [[positive number|positive]] [[multiple]] of an abundant number is an abundant number. |
| − | Every multiple of a perfect number greater than itself is an abundant number. | + | Every [[multiple]] of a [[perfect number]] greater than itself is an abundant number. |
==See also== | ==See also== | ||
Latest revision as of 11:27, 9 February 2018
An abundant number is a number 
for which the sum of its proper divisors is greater than itself. For example, 12 is abundant because the sum of its proper divisors is 1 + 2 + 3 + 4 + 6 = 16 > 12.
Examples
The first 10 abundant numbers are:
12, 18, 20, 24, 30, 36, 40, 42, 48, 54,...
The first 10 odd abundant numbers are:
945, 1575, 2205, 2835, 3465, 4095, 4725, 5355, 5775, 5985,...
Properties
There are infinitely many abundant numbers.
Every positive multiple of an abundant number is an abundant number.
Every multiple of a perfect number greater than itself is an abundant number.
