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amenable numbers
A number  $n$  is said to be amenable if there exist  $n$  numbers  $a_1,a_2,\dots,a_n$  in  $\mathbb{Z}$  such that  $\sum_k a_k = \prod_k a_k$, i.e., their sum is equal to their product.

For example, 8 is amenable because the 8 numbers {-1, -1, 1, 1, 1, 1, 2, 4} have the same sum and product.

O. P. Lossers proved that  $n$  is amenable if and only if  $n\neq4$  and it is of the form  $4k$  or  $4k+1$.

The first amenable numbers are 1, 5, 8, 9, 12, 13, 16, 17, 20, 21, 24, 25, 28, 29, 32, 33 more terms

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

a-pointer 13 101 181 + 999968341 ABA 24 32 64 + 999939200 aban 12 13 16 + 1000000000 abundant 12 20 24 + 50000000 Achilles 72 108 200 + 999939200 admirable 12 20 24 + 99999956 alt.fact. 101 4421 326981 36614981 alternating 12 16 21 + 989898989 amicable 220 284 1184 + 999728396 anti-perfect 244 285 133857 141635817 apocalyptic 157 192 220 + 30000 arithmetic 13 17 20 + 9999997 astonishing 204 216 429 + 793039833 automorphic 25 76 376 + 787109376 balanced p. 53 157 173 + 999998789 Bell 52 877 4140 + 27644437 bemirp 1061 1601 1901 + 199680881 betrothed 48 140 1648 + 991087064 binomial 20 21 28 + 999961560 brilliant 21 25 49 + 999999421 c.decagonal 61 101 281 + 999768701 c.heptagonal 148 197 253 + 999930772 c.nonagonal 28 136 253 + 999916840 c.octagonal 25 49 81 + 999887641 c.pentagonal 16 76 141 + 999950001 c.square 13 25 41 + 999983921 c.triangular 64 85 109 + 999814960 cake 64 93 176 + 998152597 Canada 125 581 8549 Carmichael 561 1105 1729 + 993905641 Catalan 132 429 16796 + 477638700 Chen 13 17 29 + 99999721 compositorial 24 192 1728 + 696729600 congruent 13 20 21 + 9999997 constructible 12 16 17 + 858980352 cube 64 125 216 + 1000000000 Cullen 25 65 161 + 838860801 Cunningham 17 24 28 + 999950885 Curzon 21 29 33 + 199999973 cyclic 13 17 29 + 9999997 D-number 21 33 57 + 7043133 d-powerful 24 89 132 + 9999865 de Polignac 149 337 373 + 99999937 decagonal 52 85 232 + 999650497 deceptive 481 1729 2821 + 999888709 deficient 13 16 17 + 9999997 dig.balanced 12 21 37 + 199999936 double fact. 48 105 384 + 185794560 droll 72 240 672 + 995328000 Duffinian 16 21 25 + 9999997 eban 32 36 40 + 66066064 economical 13 16 17 + 20000000 emirp 13 17 37 + 199999777 emirpimes 49 85 93 + 99999997 enlightened 256 2048 2176 + 761743661 equidigital 13 16 17 + 19999985 eRAP 20 24 1104 + 999393125 esthetic 12 21 32 + 989898989 Eulerian 57 120 1013 + 848090912 evil 12 17 20 + 999999997 factorial 24 120 720 + 479001600 fibodiv 28 61 149 + 984380344 Fibonacci 13 21 89 + 701408733 Friedman 25 121 125 + 999964 frugal 125 128 256 + 999999189 gapful 100 105 108 + 1000000000 Gilda 29 49 152 + 92078232 good prime 17 29 37 + 199968469 happy 13 28 32 + 10000000 harmonic 28 140 496 + 995248800 Harshad 12 20 21 + 999999936 heptagonal 81 112 148 + 999970000 hex 37 61 169 + 999899377 hexagonal 28 45 120 + 999916840 highly composite 12 24 36 + 735134400 hoax 84 85 136 + 99999332 Hogben 13 21 57 + 999856021 Honaker 457 1049 1301 + 999843421 house 32 652 933 + 994445684 hungry 17 144 37929 + 33662541 hyperperfect 21 28 301 + 974380921 iban 12 17 20 + 777777 iccanobiF 13 124 836 + 265429972 idoneal 12 13 16 + 1848 impolite 16 32 64 + 536870912 inconsummate 65 84 161 + 999993 insolite 11112 1122112 122121216 interprime 12 21 45 + 99999980 Jacobsthal 21 85 341 + 357913941 Jordan-Polya 12 16 24 + 995328000 junction 101 105 109 + 99999968 Kaprekar 45 297 2728 + 909090909 katadrome 20 21 32 + 987654321 Lehmer 85 133 481 + 999962185 Leyland 17 32 57 + 536871753 lonely 53 120 1340 + 325737821 Lucas 29 76 521 + 969323029 lucky 13 21 25 + 9999997 Lynch-Bell 12 24 36 + 9867312 m-pointer 61 1321 11261 + 211121213 magic 65 260 369 + 995433385 magnanimous 12 16 20 + 995955112 metadrome 12 13 16 + 123456789 modest 13 29 49 + 999998001 Moran 21 45 84 + 99999965 Motzkin 21 2188 310572 18199284 narcissistic 153 8208 54748 + 912985153 nialpdrome 20 21 32 + 1000000000 nonagonal 24 204 261 + 999474456 nude 12 24 33 + 499999968 O'Halloran 12 20 36 + 924 oban 12 13 16 + 997 octagonal 21 40 65 + 999917633 odious 13 16 21 + 1000000000 Ormiston 1913 18397 19013 + 999999113 palindromic 33 44 77 + 989999989 palprime 101 181 313 + 989969989 pancake 16 29 37 + 999961561 panconsummate 12 20 21 + 3097 pandigital 21 108 120 + 381367044 partition 56 77 101 + 952050665 pentagonal 12 92 117 + 999918232 perfect 28 496 8128 33550336 pernicious 12 13 17 + 9999997 Perrin 12 17 29 + 468557684 Pierpont 13 17 37 + 725594113 plaindrome 12 13 16 + 888888889 Poulet 341 561 645 + 998724481 power 16 25 32 + 1000000000 powerful 16 25 32 + 1000000000 practical 12 16 20 + 10000000 prim.abundant 12 20 56 + 99999956 prime 13 17 29 + 999999937 primeval 13 37 113 + 100123469 pronic 12 20 56 + 999856020 Proth 13 17 25 + 999981057 pseudoperfect 12 20 24 + 33550336 rare 65 repdigit 33 44 77 + 888888888 repfigit 28 61 197 + 251133297 repunit 13 21 40 + 999856021 Rhonda 1568 4752 5265 + 983314513 Ruth-Aaron 16 24 25 + 999857817 Saint-Exupery 60 480 780 + 997350120 Sastry 328 528 13224 + 897506928 self 20 53 64 + 999999972 self-describing 4444 224444 442244 + 88888888 semiprime 21 25 33 + 99999997 sliding 20 25 29 + 986802500 Smith 85 121 265 + 99999920 sphenic 105 165 273 + 99999969 square 16 25 36 + 999950884 star 13 37 73 + 999931141 straight-line 321 333 345 + 987654321 strobogrammatic 69 88 96 + 969986696 strong prime 17 29 37 + 99999821 subfactorial 44 265 14833 + 176214841 super Niven 12 20 24 + 1000000000 super-d 69 81 105 + 9999981 superabundant 12 24 36 + 735134400 tau 12 24 36 + 1000000000 taxicab 1729 4104 13832 + 998951616 tetrahedral 20 56 84 + 999800616 tetranacci 29 56 108 + 387559437 triangular 21 28 36 + 999961560 tribonacci 13 24 44 + 334745777 trimorphic 24 25 49 + 925781249 truncatable prime 13 17 29 + 999818353 twin 13 17 29 + 999999193 uban 12 13 16 + 1000000000 Ulam 13 16 28 + 10000241 undulating 101 121 141 + 989898989 unprimeable 200 204 208 + 10000000 untouchable 52 88 96 + 999996 upside-down 28 37 64 + 989951121 vampire 1260 6880 104260 + 96977920 wasteful 12 20 24 + 9999997 weak prime 13 61 73 + 99999941 weakly prime 294001 584141 1282529 + 998740213 weird 836 7192 7912 + 997348 Wieferich 1093 10533 14209 + 748316985 Woodall 17 80 4373 + 215233604 Zeisel 105 1729 1885 + 995762585 Zuckerman 12 24 36 + 997711344 Zumkeller 12 20 24 + 100000 zygodrome 33 44 77 + 999999988