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abundant numbers
A number  $n$  is abundant if  $\sigma(n) > 2\cdot n$, i.e., if the sum of the proper divisors of  $n$  is larger than  $n$.

For example, 12 is abundant since the sum of its proper divisors, 1 + 2 + 3 + 4 + 6 = 16, exceeds 12 itself.

There are infinite abundant numbers since, for example, all the multiples of an abundant number is abundant. It has been estimated that about 1/4 of the integers are abundant.

It is possible to obtain abundant numbers lacking any combination of prime factors, but they grow fast, as shown in the table below.

fmin number lacking f
2945
3 20
5 12
2 ⋅ 3 5391411025
2 ⋅ 5 81081
3 ⋅ 5 56
2 ⋅ 3 ⋅ 5 20169691981106018776756331

Every number greater that 991 can be expressed as the sum of abundant numbers. Two addends are sufficient for every integer greater than 20161 and for every even number greater than 46.

The smallest pair of consecutive abundant numbers is (5775, 5776). The first such triple starts at 171078830, while the smallest known quadruple, found by Bruno Mishutka, starts at 141363708067871564084949719820472453374.

The first abundant numbers are 12, 18, 20, 24, 30, 36, 40, 42, 48, 54, 56, 60, 66, 70, 72, 78, 80, 84, 88, 90, 96, 100 more terms

Below, the spiral pattern of abundant numbers up to 2500. See the page on prime numbers for an explanation and links to similar pictures.

spiral pattern of abundant numbers

Abundant numbers can also be... (you may click on names or numbers and on + to get more values)

ABA 18 24 72 + 50000000 aban 12 18 20 + 50000000 Achilles 72 108 200 + 50000000 admirable 12 20 24 + 49999974 alternating 12 18 30 + 49896945 amenable 12 20 24 + 50000000 amicable 220 1184 2620 + 49215166 apocalyptic 192 220 222 + 30000 arithmetic 20 30 42 + 9999996 astonishing 204 216 3078 + 32738728 automorphic 7109376 Bell 4140 678570 betrothed 48 140 1050 + 49217084 binomial 20 36 56 + 49995000 c.heptagonal 736 3256 7568 + 49022072 c.nonagonal 820 1540 2080 + 49765276 c.octagonal 11025 99225 245025 + 49491225 c.pentagonal 276 456 1266 + 49851726 c.triangular 460 760 2584 + 49654144 cake 42 176 378 + 49235272 Catalan 42 132 1430 + 35357670 compositorial 24 192 1728 + 43545600 congruent 20 24 30 + 9999996 constructible 12 20 24 + 44738560 cube 216 1000 1728 + 49836032 Cunningham 24 48 80 + 49999040 Curzon 18 30 54 + 49999794 D-number 4095 16695 1527435 + 6310395 d-powerful 24 132 224 + 9999752 de Polignac 1561875 2012985 2410065 + 49722435 decagonal 126 540 1242 + 49946022 dig.balanced 12 42 56 + 49999968 double fact. 48 384 945 + 34459425 droll 72 240 672 + 46080000 Duffinian 36 100 144 + 9999392 eban 30 36 40 + 50000000 economical 112 160 162 + 20000000 enlightened 2176 2304 2500 + 27294568 equidigital 112 160 162 + 19999488 eRAP 20 24 1104 + 49232136 esthetic 12 54 56 + 43454565 Eulerian 66 120 8178 + 33554406 evil 12 18 20 + 50000000 factorial 24 120 720 + 39916800 fibodiv 366 3248 5466 + 47880390 Fibonacci 144 2584 46368 + 14930352 Friedman 126 216 736 + 999964 frugal 1280 1458 1536 + 50000000 gapful 100 108 120 + 50000000 Gilda 78 220 330 + 14005576 Giuga 30 858 1722 66198 happy 70 100 176 + 10000000 harmonic 140 270 672 + 46683000 Harshad 12 18 20 + 50000000 heptagonal 18 112 342 + 49967896 hexagonal 66 120 276 + 49995000 highly composite 12 24 36 + 43243200 hoax 84 160 234 + 49999292 house 78 1716 5336 + 46996332 hungry 144 82810 iban 12 20 24 + 777774 iccanobiF 836 17354310 idoneal 12 18 24 + 1848 inconsummate 84 216 272 + 999980 insolite 11112 1122112 interprime 12 18 30 + 49999956 Jacobsthal 5592405 Jordan-Polya 12 24 36 + 49766400 junction 204 208 210 + 49999860 Kaprekar 2728 4950 5292 + 49995000 katadrome 20 30 40 + 9876540 Leyland 54 100 320 + 14352282 lonely 120 1340 1344 + 47326800 Lucas 18 5778 1860498 lucky 1575 2835 3465 + 9999825 Lynch-Bell 12 24 36 + 9867312 magic 260 870 6924 + 48668230 magnanimous 12 20 30 + 5731136 metadrome 12 18 24 + 12345678 modest 222 444 618 + 49989702 Moran 18 42 84 + 49999734 Motzkin 113634 310572 6536382 18199284 narcissistic 8208 9474 93084 4210818 nialpdrome 20 30 40 + 50000000 nonagonal 24 204 396 + 49999950 nude 12 24 36 + 49999968 O'Halloran 12 20 36 + 924 oban 12 18 20 + 996 octagonal 40 96 176 + 49980008 odious 42 56 70 + 49999998 palindromic 66 88 222 + 49988994 pancake 56 352 704 + 49945016 panconsummate 12 18 20 + 144 pandigital 78 108 114 + 47754948 partition 30 42 56 + 26543660 pentagonal 12 70 176 + 49971090 pernicious 12 18 20 + 9999980 Perrin 12 90 486 + 2968530 plaindrome 12 18 24 + 48888888 power 36 100 144 + 49984900 powerful 36 72 100 + 50000000 practical 12 18 20 + 10000000 prim.abundant 12 18 20 + 49999974 primeval 10136 primorial 30 210 2310 + 9699690 pronic 12 20 30 + 49991970 Proth 7425 49665 318465 + 47734785 pseudoperfect 12 18 20 + 1000000 repdigit 66 88 222 + 6666666 repfigit 1104 2208 2580 + 33445755 repunit 40 156 364 + 49565920 Rhonda 1568 2835 4752 + 49752836 Ruth-Aaron 24 78 104 + 49944222 Saint-Exupery 60 480 780 + 49970760 Sastry 528 13224 453288 + 22145328 self 20 42 108 + 49999980 self-describing 442244 666666 10313316 + 42272724 sliding 20 70 200 + 42500000 Smith 378 438 576 + 49999544 sphenic 30 42 66 + 49999974 square 36 100 144 + 49984900 straight-line 210 222 234 + 12345678 strobogrammatic 88 96 888 + 9990666 subfactorial 1854 133496 super Niven 12 20 24 + 50000000 super-d 318 336 348 + 9999894 superabundant 12 24 36 + 36756720 tau 12 18 24 + 49999968 taxicab 4104 13832 32832 + 49933368 tcefrep 498906 20671542 41673714 tetrahedral 20 56 84 + 49902940 tetranacci 56 108 208 + 28074040 triangular 36 66 78 + 49995000 tribonacci 24 504 3136 + 8646064 trimorphic 24 624 90624 + 12890624 uban 12 18 20 + 50000000 Ulam 18 36 48 + 10000236 undulating 252 272 282 + 48484848 unprimeable 200 204 208 + 10000000 untouchable 88 96 120 + 999996 upside-down 258 456 654 + 49991116 vampire 1260 1530 6880 + 49867236 wasteful 12 18 20 + 9999996 weird 70 836 4030 + 999670 Woodall 80 15624 5764800 Zuckerman 12 24 36 + 49787136 Zumkeller 12 20 24 + 100000 zygodrome 66 88 222 + 44999988