Abundant Numbers

Complete list of abundant numbers up to 1,000

12 18 20 24 30 36 40 42 48 54 56 60 66 70 72 78 80 84 88 90 96 100 102 104 108 112 114 120 126 132 138 140 144 150 156 160 162 168 174 176 180 186 192 196 198 200 204 208 210 216 220 222 224 228 234 240 246 252 258 260 264 270 272 276 280 282 288 294 300 304 306 308 312 318 320 324 330 336 340 342 348 350 352 354 360 364 366 368 372 378 380 384 390 392 396 400 402 408 414 416 420 426 432 438 440 444 448 450 456 460 462 464 468 474 476 480 486 490 492 498 500 504 510 516 520 522 528 532 534 540 544 546 550 552 558 560 564 570 572 576 580 582 588 594 600 606 608 612 616 618 620 624 630 636 640 642 644 648 650 654 660 666 672 678 680 684 690 696 700 702 704 708 714 720 726 728 732 736 738 740 744 748 750 756 760 762 768 770 774 780 784 786 792 798 800 804 810 812 816 820 822 828 832 834 836 840 846 852 858 860 864 868 870 876 880 882 888 894 896 900 906 910 912 918 920 924 928 930 936 940 942 945 948 952 954 960 966 968 972 978 980 984 990 992 996 1000

What are Abundant Numbers?

An abundant number (also called an excessive number) is a positive integer for which the sum of its proper divisors is greater than the number itself. The smallest abundant number is 12, because its proper divisors are 1, 2, 3, 4, and 6, and their sum is 1 + 2 + 3 + 4 + 6 = 16, which exceeds 12. The abundance of 12 is 16 − 12 = 4.

The first several abundant numbers are: 12, 18, 20, 24, 30, 36, 40, 42, 48, 54, 56, 60, 66, 70, 72, 78, 80, 84, 88, 90...

Classification of Integers

Every positive integer falls into exactly one of three categories based on the sum of its proper divisors: deficient (sum less than the number), perfect (sum equals the number), or abundant (sum exceeds the number). This classification dates back to Nicomachus of Gerasa around 100 AD and has been studied for over two millennia.

Among the first 100 positive integers, there are 76 deficient numbers, 2 perfect numbers (6 and 28), and 22 abundant numbers. As numbers grow larger, the proportion of abundant numbers increases, eventually approaching a natural density of approximately 24.76% of all positive integers.

Properties of Abundant Numbers

Every multiple of an abundant number is also abundant. Since 12 is the smallest abundant number, all multiples of 12 (24, 36, 48, ...) are abundant. Every even number greater than 46 can be written as the sum of two abundant numbers. The largest number that cannot be expressed as the sum of two abundant numbers is 20161.

Every even number greater than or equal to 12 that is a multiple of 6 is abundant. The smallest odd abundant number is 945 = 33 × 5 × 7, whose proper divisors sum to 975. Odd abundant numbers are much rarer than even ones in the lower ranges.

Abundancy Index

The abundancy index of a number n is defined as σ(n)/n, where σ(n) is the sum of all divisors of n (including n itself). A number is abundant if its abundancy index exceeds 2, perfect if it equals 2, and deficient if it is less than 2. The abundancy index provides a continuous measure of how "abundant" a number is.

Highly abundant numbers — numbers whose divisor sum exceeds that of all smaller positive integers — form an important related sequence. The first several highly abundant numbers are 1, 2, 3, 4, 6, 8, 10, 12, 16, 18, 20, 24, 30, 36, 48, 60.

Applications

Abundant numbers play a role in number theory and algebra. They appear in the study of aliquot sequences — sequences formed by repeatedly taking the sum of proper divisors. The behavior of aliquot sequences is connected to deep questions about the distribution of perfect and abundant numbers. The Erdős–Straus conjecture and related problems involve properties of divisor sums that connect to abundant numbers.