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practical numbers
A number  $n$  is called practical if all the numbers  $m < n$ can be written as the sum of distinct proper divisors of  $n$.

For example, 18 is practical because every smaller number can be written as a sum of its proper divisors, 1, 2, 3, 6, 9, like 13=1+3+9.

Steward and Sierpiński have characterized completely the set of practical numbers as follows. A number  $n1$">, whose prime factorization is  $p_1^{e_1}\cdots p_k^{e_k}$  is a practical number if and only if it is even (i.e.,  $p_1=2$) and, for every  $j=2,\dots,k$, it holds

\[ p_j \le 1 + \sigma(p_1^{e_1}\cdots p_{j-1}^{e_{j-1}})\,, \]
where  $\sigma(x)$  denotes the sum of divisors of  $x$.

The first practical numbers are 1, 2, 4, 6, 8, 12, 16, 18, 20, 24, 28, 30, 32, 36, 40, 42, 48, 54, 56, 60, 64, 66, 72, 78, 80, 84, 88, 90, 96, 100, 104, 108, 112, 120, 126, 128, 132, 140, 144, 150, 156, 160 more terms

Below, the spiral pattern of practical numbers up to  $250^2$. See the page on prime numbers for an explanation and links to similar pictures.

spiral pattern of practical numbers

Pictorial representation of remainders (mod 2, 3, ...,11) frequency. For a table of values and more details click here
remainders of practical numbers

A graph displaying how many practical numbers are multiples of the primes p from 2 to 71. In black the ideal line 1/p.
divisibility of practical numbers
Useful links Mathworld, Practical Number
Wikipedia, Practical number
OEIS, Sequence A005153
Practical numbers can also be... (you may click on names or numbers and on + to get more values)

ABA 18 24 32 + 9999392 aban 12 16 18 + 10000000 abundant 12 18 20 + 10000000 Achilles 72 108 200 + 9999392 admirable 12 20 24 + 9989312 alternating 12 16 18 + 8989890 amenable 12 16 20 + 10000000 amicable 220 1184 17296 + 9363584 apocalyptic 192 220 224 + 30000 arithmetic 20 30 42 + 9999990 astonishing 204 216 3078 + 2282148 Bell 4140 betrothed 48 140 1050 + 8829792 binomial 20 28 36 + 9965880 c.heptagonal 736 3256 7568 + 9304856 c.nonagonal 28 496 820 + 9823528 c.pentagonal 16 276 456 + 9815856 c.triangular 64 460 760 + 9696460 cake 42 64 176 + 9963072 Catalan 42 132 208012 + 2674440 compositorial 24 192 1728 + 2903040 congruent 20 24 28 + 9999990 constructible 12 16 20 + 8947712 cube 64 216 512 + 9528128 Cunningham 24 28 48 + 9991920 Curzon 18 30 54 + 9999990 d-powerful 24 132 224 + 9997668 decagonal 126 540 1242 + 9880020 deficient 16 32 64 + 8388608 dig.balanced 12 42 56 + 9999990 double fact. 48 384 3840 + 645120 droll 72 240 672 + 9461760 Duffinian 16 32 36 + 9999392 eban 30 32 36 + 6066060 economical 16 32 64 + 10000000 enlightened 256 2048 2176 + 2560000 equidigital 16 32 64 + 9999936 eRAP 20 24 1104 + 9952712 esthetic 12 32 54 + 8765456 Eulerian 66 120 14608 + 2203488 evil 12 18 20 + 10000000 factorial 24 120 720 + 3628800 fibodiv 28 3248 6496 + 2329856 Fibonacci 144 46368 832040 Friedman 126 128 216 + 995364 frugal 128 256 512 + 10000000 gapful 100 108 120 + 10000000 Gilda 78 220 330 + 5346 Giuga 30 858 1722 66198 happy 28 32 100 + 10000000 harmonic 28 140 270 + 8872200 Harshad 12 18 20 + 10000000 heptagonal 18 112 342 + 9997000 hexagonal 28 66 120 + 9961416 highly composite 12 24 36 + 8648640 hoax 84 160 234 + 9999150 house 32 78 1716 + 9867520 hungry 144 hyperperfect 28 496 8128 iban 12 20 24 + 777744 idoneal 12 16 18 + 1848 impolite 16 32 64 + 8388608 inconsummate 84 216 272 + 999978 insolite 1122112 interprime 12 18 30 + 9999792 Jordan-Polya 12 16 24 + 9953280 junction 204 208 210 + 9999858 Kaprekar 2728 4950 5292 + 8161912 katadrome 20 30 32 + 9876540 Leyland 32 54 100 + 1941760 lonely 120 1344 15704 + 2010800 Lucas 18 5778 Lynch-Bell 12 24 36 + 9867312 magic 260 870 16400 + 8001630 magnanimous 12 16 20 + 115136 metadrome 12 16 18 + 1245678 modest 666 812 888 + 9964482 Moran 18 42 84 + 888 narcissistic 8208 nialpdrome 20 30 32 + 10000000 nonagonal 24 204 396 + 9909504 nude 12 24 36 + 9999936 O'Halloran 12 20 36 + 924 oban 12 16 18 + 990 octagonal 40 96 176 + 9999176 odious 16 28 32 + 9999980 palindromic 66 88 252 + 8992998 pancake 16 56 352 + 9845704 panconsummate 12 18 20 + 144 pandigital 78 108 120 + 9998058 partition 30 42 56 + 8118264 pentagonal 12 176 210 + 9983310 perfect 28 496 8128 pernicious 12 18 20 + 9999980 Perrin 12 90 486 + 2968530 plaindrome 12 16 18 + 6678888 power 16 32 36 + 10000000 powerful 16 32 36 + 10000000 prim.abundant 12 18 20 + 9989312 primorial 30 210 2310 + 9699690 pronic 12 20 30 + 9988760 pseudoperfect 12 18 20 + 1000000 repdigit 66 88 666 + 888888 repfigit 28 1104 2208 + 694280 repunit 40 156 364 + 9984816 Rhonda 1568 4752 5664 + 9541116 Ruth-Aaron 16 24 78 + 9997416 Saint-Exupery 60 480 780 + 9982500 Sastry 528 13224 453288 2975208 self 20 42 64 + 9999840 self-describing 666666 sliding 20 200 520 + 9250000 Smith 378 576 588 + 9997840 sphenic 30 42 66 78 square 16 36 64 + 9998244 straight-line 210 234 420 + 8765432 strobogrammatic 88 96 888 + 9990666 subfactorial 133496 super Niven 12 20 24 + 10000000 super-d 336 348 462 + 9999664 superabundant 12 24 36 + 8648640 tau 12 18 24 + 10000000 taxicab 4104 13832 32832 + 9560896 tetrahedral 20 56 84 + 9962680 tetranacci 56 108 208 + 147312 triangular 28 36 66 + 9965880 tribonacci 24 504 3136 + 8646064 trimorphic 24 624 90624 + 2890624 uban 12 16 18 + 10000000 Ulam 16 18 28 + 9999960 undulating 252 272 414 + 6969696 unprimeable 200 204 208 + 10000000 untouchable 88 96 120 + 999990 upside-down 28 64 456 + 8765432 vampire 1260 1530 6880 + 829696 wasteful 12 18 20 + 9999990 Woodall 80 15624 5764800 Zuckerman 12 24 36 + 9813312 Zumkeller 12 20 24 + 100000 zygodrome 66 88 666 + 9999900