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arithmetic numbers
A number  $n$  is called arithmetic if the arithmetic mean of its divisors is an integer number.

For example, 42 is arithmetic because it has 8 divisors (1, 2, 3, 6, 7, 14, 21, 42) whose average is 96/8 = 12.

All the odd prime numbers are arithmetic. It is possible to find quite long runs of consecutive arithmetic numbers. For example, one of length 105 starts at 3033935561.

Antonio M. Oller-MarcĂ©n has proved various properties of arithmetic numbers, for example, that if  $p$  is an odd prime then  $p^p$  is arithmetic.

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

spiral pattern of arithmetic numbers

The smallest 3 × 3 magic square made of consecutive arithmetic numbers is

473946
434445
424941

The first arithmetic numbers are 1, 3, 5, 6, 7, 11, 13, 14, 15, 17, 19, 20, 21, 22, 23, 27, 29, 30, 31, 33, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 51, 53, 54, 55, 56, 57, 59, 60, 61, 62, 65, 66, 67, 68, 69, 70 more terms

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

a-pointer 11 13 101 + 9999347 ABA 375 1029 1215 + 9923847 aban 11 13 14 + 9000998 abundant 20 30 42 + 9999996 Achilles 432 500 864 + 9948825 admirable 20 30 42 + 9999942 alt.fact. 19 101 619 + 3301819 alternating 14 21 23 + 9898989 amenable 13 17 20 + 9999997 amicable 220 284 2620 + 9892936 anti-perfect 285 133857 apocalyptic 157 220 222 + 29999 astonishing 15 27 204 + 7103835 automorphic 376 90625 2890625 balanced p. 53 157 173 + 9999937 Bell 15 203 877 + 4213597 bemirp 1061 1091 1601 + 1909081 betrothed 140 195 1050 + 9345903 binomial 15 20 21 + 9997156 brilliant 14 15 21 + 9999811 c.decagonal 11 31 61 + 9989911 c.heptagonal 22 43 71 + 9978613 c.nonagonal 55 91 190 + 9997156 c.octagonal 49 169 361 + 9853321 c.pentagonal 31 51 141 + 9995001 c.square 13 41 61 + 9994921 c.triangular 19 31 46 + 9996214 cake 15 42 93 + 9886826 Canada 125 581 8549 16999 Carmichael 561 1105 1729 + 9890881 Carol 47 223 959 + 4190207 Catalan 14 42 132 + 9694845 Chen 11 13 17 + 9999991 congruent 13 14 15 + 9999999 constructible 15 17 20 + 6684774 cube 27 125 343 + 9938375 Cullen 65 161 385 + 9961473 Cunningham 15 17 31 + 9999999 Curzon 14 21 29 + 9999990 cyclic 11 13 15 + 9999997 D-number 15 21 33 + 7043133 d-powerful 43 89 132 + 9999865 de Polignac 127 149 251 + 9999997 decagonal 27 85 126 + 9993501 deceptive 91 259 451 + 9989161 deficient 11 13 14 + 9999999 dig.balanced 11 15 19 + 9999996 double fact. 15 105 945 + 645120 droll 672 14080 116736 + 7987200 Duffinian 21 27 35 + 9999997 eban 30 42 44 + 6066066 economical 11 13 14 + 9999991 emirp 13 17 31 + 9999971 emirpimes 15 39 49 + 9999998 enlightened 2744 219488 290912 5117695 equidigital 11 13 14 + 9999991 eRAP 20 2079 2255 + 9960047 esthetic 21 23 43 + 9898989 Eulerian 11 57 66 + 9738114 evil 15 17 20 + 9999999 fibodiv 14 19 47 + 9869390 Fibonacci 13 21 55 + 9227465 Friedman 125 126 127 + 999964 frugal 125 343 1029 + 9999803 gapful 105 110 132 + 9999990 Gilda 29 49 78 + 4848955 Giuga 30 858 1722 66198 good prime 11 17 29 + 9993337 happy 13 19 23 + 9999992 harmonic 140 270 672 + 8872200 Harshad 20 21 27 + 9999990 heptagonal 55 189 235 + 9977013 hex 19 37 61 + 9997351 hexagonal 15 45 66 + 9997156 highly composite 60 840 2520 + 8648640 hoax 22 85 94 + 9999895 Hogben 13 21 31 + 9995083 Honaker 131 263 457 + 9987001 house 78 155 271 + 9867520 hungry 17 2003 37929 + 7339199 hyperperfect 21 301 697 + 9699181 iban 11 14 17 + 777777 iccanobiF 13 39 836 + 9679838 idoneal 13 15 21 + 1848 inconsummate 62 65 95 + 999995 insolite 111 11112 interprime 15 21 30 + 9999982 Jacobsthal 11 21 43 + 5592405 Jordan-Polya 96 432 480 + 9953280 junction 101 103 105 + 9999961 Kaprekar 45 55 99 + 9999999 katadrome 20 21 30 + 9876543 Kynea 23 79 287 + 4198399 Lehmer 15 51 85 + 9997351 Leyland 17 54 57 + 9865625 lonely 23 53 211 + 4652430 Lucas 11 29 47 + 4870847 lucky 13 15 21 + 9999997 Lynch-Bell 15 126 132 + 9867312 m-pointer 23 61 1123 + 9111341 magic 15 65 111 + 9951391 magnanimous 11 14 20 + 9959374 metadrome 13 14 15 + 3456789 modest 13 19 23 + 9999999 Moran 21 27 42 + 9999950 Motzkin 21 51 127 + 6536382 narcissistic 153 371 407 + 9926315 nialpdrome 11 20 21 + 9999999 nonagonal 46 111 154 + 9992125 nude 11 15 22 + 9999999 O'Halloran 20 44 60 + 924 oban 11 13 15 + 999 octagonal 21 65 96 + 9999176 odious 11 13 14 + 9999998 Ormiston 1913 1931 18379 + 9994631 palindromic 11 22 33 + 9999999 palprime 11 101 131 + 9989899 pancake 11 22 29 + 9997157 panconsummate 11 14 15 + 3097 pandigital 11 15 19 + 9998170 partition 11 15 22 + 9289091 pentagonal 22 35 51 + 9998795 pernicious 11 13 14 + 9999998 Perrin 17 22 29 + 9141872 Pierpont 13 17 19 + 8503057 plaindrome 11 13 14 + 9999999 Poulet 341 561 645 + 9995671 power 27 49 125 + 9938375 powerful 27 49 125 + 9948825 practical 20 30 42 + 9999990 prim.abundant 20 30 42 + 9999942 prime 11 13 17 + 9999991 primeval 13 37 107 + 1234679 primorial 30 210 2310 + 9699690 pronic 20 30 42 + 9995082 Proth 13 17 33 + 9998337 pseudoperfect 20 30 42 + 999999 rare 65 repdigit 11 22 33 + 9999999 repfigit 14 19 47 + 7913837 repunit 13 15 21 + 9995083 Rhonda 4752 5439 5664 + 9843255 Ruth-Aaron 15 49 77 + 9997417 Saint-Exupery 60 480 780 + 9903180 Sastry 183 715 6099 + 2975208 self 20 31 42 + 9999998 self-describing 22 4444 224444 + 666666 semiprime 14 15 21 + 9999998 sliding 11 20 29 + 9868025 Smith 22 27 85 + 9999895 sphenic 30 42 66 + 9999994 square 49 169 361 + 9853321 star 13 37 73 + 9992341 straight-line 111 123 135 + 9999999 strobogrammatic 11 69 96 + 9998666 strong prime 11 17 29 + 9999971 subfactorial 44 265 1854 14833 super Niven 20 30 60 + 9909000 super-d 19 31 69 + 9999981 superabundant 60 840 2520 + 8648640 tau 56 60 96 + 9999992 taxicab 1729 4104 13832 + 9443761 tcefrep 498906 tetrahedral 20 35 56 + 9962680 tetranacci 15 29 56 + 7555935 triangular 15 21 45 + 9997156 tribonacci 13 44 149 + 8646064 trimorphic 49 51 99 + 9999999 truncatable prime 13 17 23 + 9986113 twin 11 13 17 + 9999973 uban 11 13 15 + 9000099 Ulam 11 13 38 + 9999999 undulating 101 131 141 + 9898989 unprimeable 204 206 322 + 9999986 untouchable 96 188 206 + 999996 upside-down 19 37 46 + 9995111 vampire 1395 1435 1827 + 841995 wasteful 20 22 30 + 9999999 weak prime 13 19 23 + 9999991 weakly prime 294001 505447 584141 + 9931447 weird 70 836 4030 + 999670 Wieferich 1093 3279 3511 + 6161805 Woodall 17 23 159 + 9961471 Zeisel 105 1419 1729 + 9773731 Zuckerman 11 15 111 + 9618912 Zumkeller 20 30 42 + 99996 zygodrome 11 22 33 + 9999999