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Term

Abundant Number

Numbers whose proper divisors sum to more than the number itself.

521 numbers tagged.

A positive integer is abundant when the sum of its proper divisors exceeds the number itself. For example, 12's proper divisors are 1, 2, 3, 4, 6 — summing to 16, more than 12. So 12 is abundant.

The smallest abundant number is 12. The first several: 12, 18, 20, 24, 30, 36, 40, 42, 48. All multiples of an abundant number are also abundant. Every multiple of 6 greater than 6 is abundant.

Abundant numbers are far more common than perfect numbers but less common than deficient numbers. The density of abundant numbers in the natural numbers is approximately 0.2476.

1,638 1,640 1,644 1,650 1,652 1,656 1,660 1,662 1,664 1,668 1,672 1,674 1,680 1,686 1,692 1,696 1,698 1,700 1,704 1,708 1,710 1,716 1,720 1,722 1,728 1,734 1,736 1,740 1,746 1,750 1,752 1,758 1,760 1,764 1,768 1,770 1,776 1,780 1,782 1,788 1,792 1,794 1,800 1,806 1,812 1,818 1,820 1,824 1,830 1,836 1,840 1,842 1,848 1,854 1,856 1,860 1,866 1,870 1,872 1,876 1,878 1,880 1,884 1,888 1,890 1,896 1,900 1,902 1,904 1,908 1,914 1,920 1,926 1,932 1,936 1,938 1,940 1,944 1,950 1,952 1,956 1,960 1,962 1,968 1,974 1,976 1,980 1,984 1,986 1,988 1,992 1,998 2,000 2,002 2,004 2,010 2,016 2,020 2,022 2,024 2,028 2,030 2,034 2,040 2,044 2,046 2,052 2,058 2,060 2,064 2,070 2,072 2,076 2,080 2,082 2,088 2,090 2,094 2,100 86,400 31,536,000