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pseudoperfect numbers
A number  $n$  is called pseudoperfect if it is equal to the sum of a subset of its proper divisors.

For example, 12 is pseudoperfect since it can be written as 2+4+6.

Pseudoperfect numbers are also called semiperfect.

All pseudoperfect numbers are clearly perfect or abundant and all the practical numbers, except powers of 2, are also pseudoperfect.

The multiples of a pseudoperfect number are pseudoperfect and a number of the form  $2^k\cdot m$  is surely pseudoperfect if there is a prime  $p<2^{k+1}$  which divides  $m$.

According to Wenjie Fang, all the odd abundant number below  $1.8\cdot10^{19}$  are also pseudoperfect.

An abundant number which is not pseudoperfect is called weird.

The first pseudoperfect numbers are 6, 12, 18, 20, 24, 28, 30, 36, 40, 42, 48, 54, 56, 60, 66, 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 more terms

Below, the spiral pattern of pseudoperfect numbers up to  $100^2$. See the page on prime numbers for an explanation and links to similar pictures.

spiral pattern of pseudoperfect numbers

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

ABA 18 24 72 + 994050 aban 12 18 20 + 1000000 abundant 12 18 20 + 1000000 Achilles 72 108 200 + 998784 admirable 12 20 24 + 999942 alternating 12 18 30 + 989898 amenable 12 20 24 + 33550336 amicable 220 1184 2620 + 998104 apocalyptic 192 220 222 + 30000 arithmetic 20 30 42 + 999999 astonishing 204 216 3078 + 23490 Bell 4140 678570 betrothed 48 140 1050 + 587460 binomial 20 28 36 + 33550336 c.heptagonal 736 3256 7568 + 866272 c.nonagonal 28 496 820 + 33550336 c.octagonal 11025 99225 245025 + 893025 c.pentagonal 276 456 1266 + 978126 c.triangular 460 760 2584 + 975664 cake 42 176 378 + 956040 Catalan 42 132 1430 + 742900 compositorial 24 192 1728 + 207360 congruent 20 24 28 + 999999 constructible 12 20 24 + 4294901760 cube 216 1000 1728 + 1000000 Cunningham 24 28 48 + 999999 Curzon 18 30 54 + 999978 D-number 4095 16695 d-powerful 24 132 224 + 999252 decagonal 126 540 1242 + 998500 dig.balanced 12 42 56 + 999996 double fact. 48 384 945 + 645120 droll 72 240 672 + 718848 Duffinian 36 100 144 + 1000000 eban 30 36 40 + 66066 economical 112 160 162 + 1000000 enlightened 2176 2304 2500 + 256000 equidigital 112 160 162 + 998816 eRAP 20 24 1104 + 991248 esthetic 12 54 56 + 989898 Eulerian 66 120 8178 + 524268 evil 12 18 20 + 999999 factorial 24 120 720 + 362880 fibodiv 28 366 3248 + 974994 Fibonacci 144 2584 46368 832040 Friedman 126 216 736 + 999964 frugal 1280 1458 1536 + 33550336 gapful 100 108 120 + 4294901760 Gilda 78 220 330 + 933138 Giuga 30 858 1722 66198 happy 28 100 176 + 1000000 harmonic 28 140 270 + 33550336 Harshad 12 18 20 + 999990 heptagonal 18 112 342 + 997612 hexagonal 28 66 120 + 33550336 highly composite 12 24 36 + 720720 hoax 84 160 234 + 999860 house 78 1716 5336 + 758952 hungry 144 82810 hyperperfect 28 496 8128 33550336 iban 12 20 24 + 777774 idoneal 12 18 24 + 1848 inconsummate 84 216 272 + 999980 insolite 11112 interprime 12 18 30 + 999960 Jordan-Polya 12 24 36 + 995328 junction 204 208 210 + 999950 Kaprekar 2728 4950 5292 + 999999 katadrome 20 30 40 + 987654 Leyland 54 100 320 + 94932 lonely 120 1340 1344 + 492170 Lucas 18 5778 lucky 1575 2835 3465 + 999075 Lynch-Bell 12 24 36 + 984312 magic 260 870 6924 + 864060 magnanimous 12 20 30 + 975772 metadrome 12 18 24 + 345678 modest 222 444 618 + 999999 Moran 18 42 84 + 999612 Motzkin 113634 310572 narcissistic 8208 9474 93084 nialpdrome 20 30 40 + 1000000 nonagonal 24 204 396 + 985536 nude 12 24 36 + 999999 O'Halloran 12 20 36 + 924 oban 12 18 20 + 996 octagonal 40 96 176 + 994176 odious 28 42 56 + 33550336 palindromic 66 88 222 + 999999 pancake 56 352 704 + 998992 panconsummate 12 18 20 + 144 pandigital 78 108 114 + 797895 partition 30 42 56 + 831820 pentagonal 12 176 210 + 998376 perfect 28 496 8128 33550336 pernicious 12 18 20 + 1000000 Perrin 12 90 486 + 76725 plaindrome 12 18 24 + 999999 power 36 100 144 + 1000000 powerful 36 72 100 + 1000000 practical 12 18 20 + 1000000 prim.abundant 12 18 20 + 999999 primeval 10136 primorial 30 210 2310 + 510510 pronic 12 20 30 + 4294901760 Proth 7425 49665 318465 + 963585 repdigit 66 88 222 + 999999 repfigit 28 1104 2208 + 694280 repunit 40 156 364 + 980200 Rhonda 1568 2835 4752 + 625275 Ruth-Aaron 24 78 104 + 999072 Saint-Exupery 60 480 780 + 994560 Sastry 528 13224 453288 self 20 42 108 + 1000000 self-describing 442244 666666 sliding 20 200 520 + 925000 Smith 378 438 576 + 999950 sphenic 30 42 66 + 999942 square 36 100 144 + 1000000 straight-line 210 222 234 + 999999 strobogrammatic 88 96 888 + 999666 subfactorial 1854 133496 super Niven 12 20 24 + 1000000 super-d 318 336 348 + 999894 superabundant 12 24 36 + 720720 tau 12 18 24 + 999920 taxicab 4104 13832 32832 + 994688 tcefrep 498906 tetrahedral 20 56 84 + 988260 tetranacci 56 108 208 + 147312 triangular 28 36 66 + 33550336 tribonacci 24 504 3136 + 66012 trimorphic 24 624 90624 + 999999 uban 12 18 20 + 1000000 Ulam 18 28 36 + 1000000 undulating 252 272 282 + 989898 unprimeable 200 204 208 + 999996 untouchable 88 96 120 + 999996 upside-down 28 258 456 + 899112 vampire 1260 1530 6880 + 841995 wasteful 12 18 20 + 999999 Woodall 80 15624 Zuckerman 12 24 36 + 973728 Zumkeller 12 20 24 + 100000 zygodrome 66 88 222 + 999999