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Anti-perfetti (numeri)

Rappresentazione dei numeri 

Si chiamano “anti-perfetti” i numeri naturali uguali alla somma dei loro divisori, escluso il numero stesso, scritti a rovescio. Per esempio, 244 è anti-perfetto, perché i suoi divisori, escludendo 244, sono: 1, 2, 4, 61 e 122 e 1 + 2 + 4 + 16 + 221 = 244.

 

I numeri anti-perfetti noti sono: 6, 244, 285, 133857, 141635817, 273722772124 (Joseph L. Pe, Farideh Firoozbakht e Giovanni Resta, The Online Encyclopedia of Integer Sequences http://oeis.org).

 

E’ plausibile che i numeri anti-perfetti siano infiniti; i successivi in base 10, se esistono, sono maggiori di 1012 (Giovanni Resta, 2017).

 

La definizione si può estendere ad altre basi; la tabella seguente mostra i numeri anti-perfetti fino a 109 nelle basi da 2 a 20 (M. Fiorentini, 2017).

Base

Numeri anti-perfetti

2

1000000101000001001111012 = 8470845

3

1013 = 10, 2113 = 22, 10103 = 30, 20113 = 58, 22113 = 76, 102103 = 102, 200113 = 166, 1022103 = 318, 1100113 = 328, 1100203 = 330, 11001103 = 984, 11022113 = 1048, 110202203 = 3102, 111012113 = 3208, 202101113 = 4954, 202112023 = 4988, 1000120203 = 6702, 2000000113 = 13126, 2021120203 = 14964, 2021201113 = 14998, 2212220213 = 18934, 10222222103 = 26238, 11221222103 = 32556, 20021010113 = 41098, 21221200113 = 52168, 101022120103 = 67692, 101200102103 = 70086, 102121022103 = 76134, 102222222103 = 78726, 120100112023 = 100730, 200000000113 = 118102, 202122201113 = 135526, 1100010120113 = 237064, 1110222212113 = 262408, 1120001121103 = 275952, 2002122010113 = 371578, 2210100210213 = 494458, 2212222220213 = 511738, 10002222120203 = 551022, 10212001022103 = 682662, 11102222212113 = 787288, 11220212022103 = 871716, 22101000210213 = 1482982, 100012010120203 = 1627998, 101200101021213 = 1892068, 200002101000113 = 3204202, 200210121010113 = 3330346, 200212222010113 = 3345898, 221012101210213 = 4464106, 1000212102120203 = 4936686, 1000222222120203 = 4960014, 1010221012120103 = 5479788, 1022222222222103 = 6377286, 2000000000000113 = 9565942, 2021222222201113 = 10983046, 2212201021220213 = 13766902, 10120000000102103 = 17006214, 10121022212102103 = 17241438, 10212000001022103 = 18423606, 10212102121022103 = 18499422, 10221200010222103 = 18896646, 10222120102222103 = 19056054, 11102222222212113 = 21257608, 20000000000000113 = 28697818, 20021222222010113 = 30114778, 20210121012101113 = 32524186, 20212222222201113 = 32949274, 22101000000210213 = 40035418, 22101021010210213 = 40082074, 22111002011101103 = 40580850, 22120012001200213 = 40954174, 100012000010120203 = 43933326, 100020022120220203 = 44165958, 101200000000102103 = 51018438, 101210222212102103 = 51726054, 102212000010222103 = 56688006, 102221200102222103 = 57162342, 110112200222121113 = 60000250, 111022222222212113 = 63772888, 200002122201000113 = 86563894, 201110020011210113 = 93042706, 202101212012101113 = 97583062, 202122010212201113 = 98696326, 220002121102212203 = 115253412, 221201212012120213 = 123092278, 221222010212220213 = 124205542, 221222222222220213 = 124357174, 1012101201012102103 = 154952454, 1021201021201022103 = 166037358, 1022121021210222103 = 170289534, 2002120100212010113 = 270568138, 2021201210121201113 = 295268746, 2220110102220121013 = 375275494, 10001200000010120203 = 395392974, 10002120000102120203 = 399646446, 10002212001022120203 = 401069454, 10002221210222120203 = 401559342, 10212000000001022103 = 497429094, 10221201012010222103 = 510394278, 11000101222010120113 = 518503576, 20000021222010000113 = 776252326, 20021201000212010113 = 811693174, 20021220102122010113 = 812649622, 20210121222012101113 = 878289574, 20212012120121201113 = 885837478, 20212201000212201113 = 888220822

4

124 = 6, 1304 = 28, 3104 = 52, 31114 = 213, 202204 = 552, 1000103014 = 65841, 2021302204 = 141096, 2022322204 = 142248, 31312312114 = 908133, 31312312312114 = 58121061, 2022303030322204 = 582169512

5

115 = 6, 245 = 14, 20045 = 254, 21305 = 290, 23145 = 334, 111105 = 780, 231025 = 1652, 1201405 = 4420, 2344125 = 8732, 3304445 = 11374, 22444025 = 40602, 213131305 = 182290, 234444145 = 218734, 12003134045 = 2744854, 23131313145 = 5208334, 42221033345 = 8784844, 1103342312405 = 60070820, 1140001440105 = 66412380, 1401142413215 = 88430836, 2003131310045 = 98958254, 2003413410045 = 99168254, 2344444444145 = 136718734, 32220020330015 = 853549126

6

106 = 6, 446 = 28, 1526 = 68, 53116 = 1195, 141526 = 2228, 1403306 = 13086, 4323246 = 35548, 14141526 = 79988, 15124236 = 87423, 432323246 = 1279708, 545413236 = 1630707, 53512534116 = 56896495

7

67 = 6, 457 = 33, 11267 = 412, 35437 = 1305, 5333147 = 92425, 5440027 = 95013, 35412307 = 447048, 35462407 = 448770, 345113107 = 3028158, 1123265267 = 6881188, 1131520447 = 6970821, 244264350217 = 751853271

8

68 = 6, 768 = 62, 2458 = 165, 7608 = 496, 25258 = 1365, 72568 = 3758, 1343358 = 47325, 7777768 = 262142, 72057720568 = 974648366, 72525252568 = 984263342

9

69 = 6, 249 = 22, 1479 = 124, 2049 = 166, 11709 = 873, 57049 = 4216, 200049 = 13126, 858279 = 56806, 1171709 = 70857, 1300179 = 78748, 2000049 = 118102, 2652349 = 161302, 8588279 = 511702, 26582349 = 1456006, 42840609 = 2299320, 43761309 = 2353320, 117171709 = 5739561, 200000049 = 9565942, 282608609 = 13975632, 316766309 = 15285456, 452631349 = 21948844, 858888279 = 41452342, 2065823049 = 89631526, 6073228089 = 262192796,

10

6, 244, 285, 133857, 141635817

11

611 = 6, 2411 = 26, 8711 = 95, 9711 = 106, 11311 = 135, 12511 = 148, 111511 = 1468, 293411 = 3788, 9AA711 = 13306, 14663011 = 228360, 20000411 = 322106, 27373411 = 429466, 39794411 = 625376, 256646611 = 4444765, 287127811 = 4935675, 983067211 = 17277185, 937AA37711 = 181986746, 9750616111 = 188598895

12

612 = 6, B91112 = 20317, 79857212 = 1943078, 2372A62312 = 82423035

13

613 = 6, 2213 = 28, 3613 = 45, 10313 = 172, 268213 = 5514, 101B313 = 28876, 1524C13 = 39948, 157BC13 = 40884, 26ACA213 = 938082, C0001313 = 4455532, 101B1B313 = 4879852, 15A152C13 = 6971964, 15A6C2C13 = 6984132, 163149C13 = 7143252, 2AB278213 = 13686402, 15504BA013 = 88749804, 1B4CB18013 = 117695760, A41C702613 = 647521842, 102BB2BB313 = 829789132

14

614 = 6, 1A5614 = 4780, 90C814 = 24872, B49814 = 31102, CD9D14 = 35615, 1467814 = 50674, 1B60A414 = 977008, 90D0C814 = 4876264, BD49D814 = 6428402, 5A31A7D114 = 604043763

15

615 = 6, 1D15 = 28, 2415 = 34, 7615 = 111, 13215 = 272, 13315 = 273, 20415 = 454, 95C815 = 31688, 2000415 = 101254, 20000415 = 1518754, 2E8A6215 = 2256842, 8E8A6D415 = 102196549, 20A9450415 = 349782754

16

616 = 6, 1C16 = 28, 159F16 = 5535, 175516 = 5973, 23D812D16 = 37585197

17

617 = 6, 1B17 = 28, 2417 = 38, 17217 = 410, 123617 = 5548, 2B5417 = 13094, 2BGG5417 = 3841766, D500B917 = 18875942, 1FD3D4917 = 46539770, 40642G417 = 97071908

18

618 = 6, 1A18 = 28, GH0818 = 98828, 154GB3818 = 43976906

19

619 = 6, 1919 = 28, CG19 = 244, 15319 = 459, 3I619 = 1431, C0G19 = 4348, 146419 = 8421, 173C419 = 179649, C000G19 = 1563868, 14ICC419 = 3125409, 31B09319 = 7634241, C0000G19 = 29713204, 16I24A419 = 64263609

20

620 = 6, 1820 = 28, 45C6620 = 684926, 134FHB20 = 3718351, EBGE3620 = 46693666, 972A72720 = 598802847, 97H9C2720 = 601196847

 

Tutti i numeri perfetti sono anti-perfetti nelle basi maggiori del loro massimo divisore proprio; per esempio, 6 è anti-perfetto in tutte le basi maggiori di 3.

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