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Automorfi esponenziali (numeri)

Rappresentazione dei numeri 

Una variante sul tema dei numeri automorfi è costituita dai numeri naturali n, tali che nn termini con le stesse cifre di n. Per esempio, 2121 = 5842587018385982521381124421 termina con 21.

 

Sono infiniti in qualsiasi base, perché in ogni base b comprendono:

  • i numeri automorfi;

  • i numeri trimorfi dispari;

  • per ogni intero n, maggiore di 1, i numeri formati da n cifre qualsiasi, seguite da n – 1 zeri e da un 1; per esempio, in base 10 sono automorfi esponenziali: 91, 1201, 345001, 87650001 ecc..

Inoltre se la base b è pari, bn – 1 è automorfo esponenziale per ogni intero positivo n.

 

Un numero automorfo esponenziale in una base resta tale nelle potenze della base che hanno per esponente una potenza di 2. Per esempio, un numero automorfo esponenziale in base 2 resta tale in base 22 = 4, 24 = 16, 28 = 256 ecc..

 

La tabella seguente riporta i numeri automorfi esponenziali fino a 1000 nelle basi fino a 20.

Base

Numeri automorfi esponenziali

2

12 = 1, 112 = 3, 1012 = 5, 1112 = 7, 10012 = 9, 11012 = 13, 11112 = 15, 100012 = 17, 110012 = 25, 111112 = 31, 1000012 = 33, 1010012 = 41, 1100012 = 49, 1110012 = 57, 1111112 = 63, 10000012 = 65, 10100012 = 81, 11000012 = 97, 11100012 = 113, 11111112 = 127, 100000012 = 129, 100100012 = 145, 101000012 = 161, 101100012 = 177, 110000012 = 193, 110100012 = 209, 111000012 = 225, 111100012 = 241, 111111112 = 255, 1000000012 = 257, 1001000012 = 289, 1010000012 = 321, 1011000012 = 353, 1100000012 = 385, 1101000012 = 417, 1110000012 = 449, 1111000012 = 481, 1111111112 = 511, 10000000012 = 513, 10001000012 = 545, 10010000012 = 577, 10011000012 = 609, 10100000012 = 641, 10101000012 = 673, 10110000012 = 705, 10111000012 = 737, 11000000012 = 769, 11001000012 = 801, 11010000012 = 833, 11011000012 = 865, 11100000012 = 897, 11101000012 = 929, 11110000012 = 961, 11111000012 = 993

3

13 = 1, 113 = 4, 213 = 7, 1013 = 10, 2013 = 19, 10013 = 28, 11013 = 37, 12013 = 46, 20013 = 55, 21013 = 64, 22013 = 73, 100013 = 82, 110013 = 109, 120013 = 136, 200013 = 163, 210013 = 190, 220013 = 217, 1000013 = 244, 1010013 = 271, 1020013 = 298, 1100013 = 325, 1110013 = 352, 1120013 = 379, 1200013 = 406, 1210013 = 433, 1220013 = 460, 2000013 = 487, 2010013 = 514, 2020013 = 541, 2100013 = 568, 2110013 = 595, 2120013 = 622, 2200013 = 649, 2210013 = 676, 2220013 = 703, 10000013 = 730, 10100013 = 811, 10200013 = 892, 11000013 = 973

4

14 = 1, 34 = 3, 114 = 5, 134 = 7, 214 = 9, 314 = 13, 334 = 15, 1014 = 17, 1214 = 25, 1334 = 31, 2014 = 33, 2214 = 41, 3014 = 49, 3214 = 57, 3334 = 63, 10014 = 65, 11014 = 81, 12014 = 97, 13014 = 113, 13334 = 127, 20014 = 129, 21014 = 145, 22014 = 161, 23014 = 177, 30014 = 193, 31014 = 209, 32014 = 225, 33014 = 241, 33334 = 255, 100014 = 257, 102014 = 289, 110014 = 321, 112014 = 353, 120014 = 385, 122014 = 417, 130014 = 449, 132014 = 481, 133334 = 511, 200014 = 513, 202014 = 545, 210014 = 577, 212014 = 609, 220014 = 641, 222014 = 673, 230014 = 705, 232014 = 737, 300014 = 769, 302014 = 801, 310014 = 833, 312014 = 865, 320014 = 897, 322014 = 929, 330014 = 961, 332014 = 993

5

15 = 1, 115 = 6, 215 = 11, 315 = 16, 415 = 21, 1015 = 26, 2015 = 51, 2125 = 57, 3015 = 76, 4015 = 101, 10015 = 126, 11015 = 151, 12015 = 176, 13015 = 201, 14015 = 226, 20015 = 251, 21015 = 276, 22015 = 301, 23015 = 326, 24015 = 351, 30015 = 376, 31015 = 401, 32015 = 426, 33015 = 451, 34015 = 476, 40015 = 501, 41015 = 526, 42015 = 551, 43015 = 576, 44015 = 601, 100015 = 626, 110015 = 751, 120015 = 876

6

16 = 1, 36 = 3, 46 = 4, 56 = 5, 116 = 7, 136 = 9, 216 = 13, 246 = 16, 256 = 17, 316 = 19, 416 = 25, 436 = 27, 446 = 28, 516 = 31, 556 = 35, 1016 = 37, 1256 = 53, 1316 = 55, 1446 = 64, 2016 = 73, 2136 = 81, 2316 = 91, 2556 = 107, 3016 = 109, 3316 = 127, 3436 = 135, 3446 = 136, 4016 = 145, 4256 = 161, 4316 = 163, 5016 = 181, 5136 = 189, 5316 = 199, 5446 = 208, 5556 = 215, 10016 = 217, 11016 = 253, 11316 = 271, 12016 = 289, 13016 = 325, 13316 = 343, 13446 = 352, 14016 = 361, 15016 = 397, 15136 = 405, 15316 = 415, 20016 = 433, 21016 = 469, 21256 = 485, 21316 = 487, 21446 = 496, 22016 = 505, 23016 = 541, 23316 = 559, 23436 = 567, 24016 = 577, 25016 = 613, 25316 = 631, 25446 = 640, 25556 = 647, 30016 = 649, 31016 = 685, 31316 = 703, 32016 = 721, 32136 = 729, 33016 = 757, 33316 = 775, 33446 = 784, 34016 = 793, 34256 = 809, 35016 = 829, 35316 = 847, 40016 = 865, 41016 = 901, 41316 = 919, 41446 = 928, 42016 = 937, 43016 = 973, 43316 = 991

7

17 = 1, 47 = 4, 117 = 8, 217 = 15, 257 = 19, 317 = 22, 417 = 29, 437 = 31, 517 = 36, 617 = 43, 1017 = 50, 2017 = 99, 3017 = 148, 4017 = 197, 5017 = 246, 6017 = 295, 6437 = 325, 10017 = 344, 11017 = 393, 12017 = 442, 13017 = 491, 14017 = 540, 15017 = 589, 16017 = 638, 20017 = 687, 21017 = 736, 22017 = 785, 23017 = 834, 24017 = 883, 25017 = 932, 26017 = 981

8

18 = 1, 38 = 3, 58 = 5, 78 = 7, 118 = 9, 218 = 17, 318 = 25, 378 = 31, 418 = 33, 518 = 41, 618 = 49, 718 = 57, 778 = 63, 1018 = 65, 1418 = 97, 2018 = 129, 2418 = 161, 3018 = 193, 3418 = 225, 3778 = 255, 4018 = 257, 4418 = 289, 5018 = 321, 5418 = 353, 6018 = 385, 6418 = 417, 7018 = 449, 7418 = 481, 7778 = 511, 10018 = 513, 11018 = 577, 12018 = 641, 13018 = 705, 14018 = 769, 15018 = 833, 16018 = 897, 17018 = 961

9

19 = 1, 49 = 4, 79 = 7, 119 = 10, 219 = 19, 319 = 28, 419 = 37, 519 = 46, 619 = 55, 719 = 64, 819 = 73, 1019 = 82, 1319 = 109, 1619 = 136, 2019 = 163, 2319 = 190, 2619 = 217, 3019 = 244, 3319 = 271, 3619 = 298, 4019 = 325, 4319 = 352, 4619 = 379, 5019 = 406, 5319 = 433, 5619 = 460, 6019 = 487, 6319 = 514, 6619 = 541, 7019 = 568, 7319 = 595, 7619 = 622, 8019 = 649, 8319 = 676, 8619 = 703, 10019 = 730, 11019 = 811, 12019 = 892, 13019 = 973

10

1, 5, 6, 9, 11, 16, 21, 25, 31, 36, 41, 49, 51, 56, 57, 61, 71, 75, 76, 81, 91, 93, 96, 99, 101, 125, 151, 176, 193, 201, 249, 251, 301, 351, 375, 376, 401, 451, 499, 501, 551, 557, 576, 601, 625, 651, 693, 701, 749, 751, 776, 801, 851, 875, 901, 951, 976, 999

11

111 = 1, 1111 = 12, 2111 = 23, 3111 = 34, 4111 = 45, 5111 = 56, 6111 = 67, 7111 = 78, 7411 = 81, 8111 = 89, 9111 = 100, A111 = 111, 10111 = 122, 13711 = 161, 20111 = 243, 30111 = 364, 40111 = 485, 50111 = 606, 60111 = 727, 70111 = 848, 80111 = 969

12

112 = 1, 312 = 3, 412 = 4, 512 = 5, 712 = 7, 912 = 9, B12 = 11, 1112 = 13, 1412 = 16, 1512 = 17, 2112 = 25, 2712 = 31, 3112 = 37, 3912 = 45, 4112 = 49, 4512 = 53, 4712 = 55, 5112 = 61, 5312 = 63, 5412 = 64, 5B12 = 71, 6112 = 73, 6712 = 79, 6912 = 81, 7112 = 85, 7512 = 89, 8112 = 97, 8712 = 103, 9112 = 109, 9412 = 112, 9912 = 117, A112 = 121, A512 = 125, A712 = 127, B112 = 133, B312 = 135, BB12 = 143, 10112 = 145, 11512 = 161, 16112 = 217, 20112 = 289, 20912 = 297, 25312 = 351, 26112 = 361, 27512 = 377, 2A712 = 415, 30112 = 433, 36112 = 505, 36912 = 513, 40112 = 577, 41512 = 593, 45412 = 640, 46112 = 649, 4A712 = 703, 50112 = 721, 50912 = 729, 56112 = 793, 57512 = 809, 5BB12 = 863, 60112 = 865, 66112 = 937, 66912 = 945, 6A712 = 991

13

113 = 1, 513 = 5, 1113 = 14, 1913 = 22, 2113 = 27, 3113 = 40, 4113 = 53, 5113 = 66, 6113 = 79, 7113 = 92, 8113 = 105, 9113 = 118, A113 = 131, B113 = 144, C113 = 157, 10113 = 170, 20113 = 339, 30113 = 508, 40113 = 677, 50113 = 846

14

114 = 1, 414 = 4, 714 = 7, 814 = 8, D14 = 13, 1114 = 15, 1514 = 19, 2114 = 29, 2314 = 31, 2814 = 36, 3114 = 43, 3714 = 49, 4114 = 57, 4814 = 64, 4B14 = 67, 5114 = 71, 5914 = 79, 6114 = 85, 6814 = 92, 6D14 = 97, 7114 = 99, 8114 = 113, 8814 = 120, 9114 = 127, A114 = 141, A714 = 147, A814 = 148, B114 = 155, C114 = 169, C814 = 176, D114 = 183, DD14 = 195, 10114 = 197, 17114 = 295, 19314 = 325, 1A714 = 343, 1A814 = 344, 1BB14 = 361, 20114 = 393, 27114 = 491, 30114 = 589, 35914 = 667, 36D14 = 685, 37114 = 687, 3A814 = 736, 40114 = 785, 47114 = 883, 50114 = 981

15

115 = 1, 515 = 5, 615 = 6, 915 = 9, A15 = 10, B15 = 11, D15 = 13, 1115 = 16, 1A15 = 25, 2115 = 31, 2615 = 36, 3115 = 46, 3415 = 49, 4115 = 61, 4B15 = 71, 5115 = 76, 5615 = 81, 6115 = 91, 6915 = 99, 6A15 = 100, 7115 = 106, 8115 = 121, 8515 = 125, 8615 = 126, 9115 = 136, A115 = 151, A715 = 157, AB15 = 161, B115 = 166, B615 = 171, BA15 = 175, C115 = 181, CD15 = 193, D115 = 196, D415 = 199, E115 = 211, E615 = 216, 10115 = 226, 18615 = 351, 20115 = 451, 30115 = 676, 31B15 = 701, 40115 = 901, 46915 = 999, 46A15 = 1000

16

116 = 1, 516 = 5, 716 = 7, 916 = 9, D16 = 13, F16 = 15, 1116 = 17, 2116 = 33, 3116 = 49, 4116 = 65, 5116 = 81, 6116 = 97, 7116 = 113, 7F16 = 127, 8116 = 129, 9116 = 145, A116 = 161, B116 = 177, C116 = 193, D116 = 209, E116 = 225, F116 = 241, FF16 = 255, 10116 = 257, 14116 = 321, 18116 = 385, 1C116 = 449, 20116 = 513, 24116 = 577, 28116 = 641, 2C116 = 705, 30116 = 769, 34116 = 833, 38116 = 897, 3C116 = 961

17

117 = 1, 917 = 9, D17 = 13, 1117 = 18, 2117 = 35, 3117 = 52, 3E17 = 65, 4117 = 69, 5117 = 86, 6117 = 103, 7117 = 120, 8117 = 137, 9117 = 154, A117 = 171, B117 = 188, C117 = 205, D117 = 222, E117 = 239, F117 = 256, G117 = 273, 10117 = 290, 20117 = 579, 30117 = 868

18

118 = 1, 418 = 4, 718 = 7, 918 = 9, A18 = 10, D18 = 13, G18 = 16, H18 = 17, 1118 = 19, 1A18 = 28, 2118 = 37, 3118 = 55, 3A18 = 64, 4118 = 73, 4918 = 81, 5118 = 91, 5A18 = 100, 6118 = 109, 7118 = 127, 7A18 = 136, 8118 = 145, 8H18 = 161, 9118 = 163, 9A18 = 172, A118 = 181, B118 = 199, BA18 = 208, C118 = 217, D118 = 235, D918 = 243, DA18 = 244, E118 = 253, F118 = 271, FA18 = 280, G118 = 289, H118 = 307, HA18 = 316, HH18 = 323, 10118 = 325, 11A18 = 352, 13118 = 379, 16118 = 433, 19118 = 487, 1C118 = 541, 1DA18 = 568, 1F118 = 595, 20118 = 649, 23118 = 703, 24918 = 729, 26118 = 757, 27A18 = 784, 29118 = 811, 2C118 = 865, 2F118 = 919, 30118 = 973, 31A18 = 1000

19

119 = 1, 719 = 7, 1119 = 20, 1919 = 28, 2119 = 39, 3119 = 58, 4119 = 77, 5119 = 96, 6119 = 115, 6D19 = 127, 7119 = 134, 8119 = 153, 9119 = 172, A119 = 191, B119 = 210, C119 = 229, D119 = 248, E119 = 267, F119 = 286, F719 = 292, G119 = 305, G319 = 307, H119 = 324, I119 = 343, 10119 = 362, 20119 = 723

20

120 = 1, 520 = 5, 920 = 9, B20 = 11, D20 = 13, F20 = 15, G20 = 16, H20 = 17, J20 = 19, 1120 = 21, 1520 = 25, 1B20 = 31, 2120 = 41, 2920 = 49, 2H20 = 57, 3120 = 61, 3B20 = 71, 4120 = 81, 4D20 = 93, 4G20 = 96, 5120 = 101, 5B20 = 111, 6120 = 121, 6520 = 125, 7120 = 141, 7920 = 149, 7B20 = 151, 7H20 = 157, 8120 = 161, 8F20 = 175, 8G20 = 176, 9120 = 181, 9B20 = 191, 9D20 = 193, 9J20 = 199, A120 = 201, B120 = 221, B520 = 225, BB20 = 231, C120 = 241, C920 = 249, CG20 = 256, CH20 = 257, D120 = 261, DB20 = 271, E120 = 281, ED20 = 293, F120 = 301, FB20 = 311, G120 = 321, G520 = 325, GG20 = 336, H120 = 341, H920 = 349, HB20 = 351, HH20 = 357, I120 = 361, IF20 = 375, J120 = 381, JB20 = 391, JD20 = 393, JJ20 = 399, 10120 = 401, 18G20 = 576, 1A120 = 601, 1B520 = 625, 20120 = 801

 

 

La tabella seguente riporta i numeri automorfi esponenziali primi fino a 1000 nelle basi fino a 20.

Base

Numeri automorfi esponenziali primi

2

112 = 3, 1012 = 5, 1112 = 7, 11012 = 13, 100012 = 17, 111112 = 31, 1010012 = 41, 11000012 = 97, 11100012 = 113, 11111112 = 127, 110000012 = 193, 111100012 = 241, 1000000012 = 257, 1011000012 = 353, 1110000012 = 449, 10010000012 = 577, 10100000012 = 641, 10101000012 = 673, 11000000012 = 769, 11101000012 = 929

3

213 = 7, 2013 = 19, 11013 = 37, 22013 = 73, 110013 = 109, 200013 = 163, 1010013 = 271, 1120013 = 379, 1210013 = 433, 2000013 = 487, 2020013 = 541, 10100013 = 811

4

34 = 3, 114 = 5, 134 = 7, 314 = 13, 1014 = 17, 1334 = 31, 2214 = 41, 12014 = 97, 13014 = 113, 13334 = 127, 30014 = 193, 33014 = 241, 100014 = 257, 112014 = 353, 130014 = 449, 210014 = 577, 220014 = 641, 222014 = 673, 300014 = 769, 322014 = 929

5

215 = 11, 4015 = 101, 11015 = 151, 20015 = 251, 31015 = 401, 44015 = 601, 110015 = 751

6

36 = 3, 56 = 5, 116 = 7, 216 = 13, 256 = 17, 316 = 19, 516 = 31, 1016 = 37, 1256 = 53, 2016 = 73, 2556 = 107, 3016 = 109, 3316 = 127, 4316 = 163, 5016 = 181, 5316 = 199, 11316 = 271, 15016 = 397, 20016 = 433, 21316 = 487, 23016 = 541, 24016 = 577, 25016 = 613, 25316 = 631, 25556 = 647, 33016 = 757, 34256 = 809, 35016 = 829, 41316 = 919, 42016 = 937, 43316 = 991

7

257 = 19, 417 = 29, 437 = 31, 617 = 43, 4017 = 197, 13017 = 491, 24017 = 883

8

38 = 3, 58 = 5, 78 = 7, 218 = 17, 378 = 31, 518 = 41, 1418 = 97, 3018 = 193, 4018 = 257, 5418 = 353, 7018 = 449, 11018 = 577, 12018 = 641, 14018 = 769

9

79 = 7, 219 = 19, 419 = 37, 819 = 73, 1319 = 109, 2019 = 163, 3319 = 271, 4619 = 379, 5319 = 433, 6019 = 487, 6619 = 541, 11019 = 811

10

5, 11, 31, 41, 61, 71, 101, 151, 193, 251, 401, 499, 557, 601, 701, 751

11

2111 = 23, 6111 = 67, 8111 = 89, 60111 = 727

12

312 = 3, 512 = 5, 712 = 7, B12 = 11, 1112 = 13, 1512 = 17, 2712 = 31, 3112 = 37, 4512 = 53, 5112 = 61, 5B12 = 71, 6112 = 73, 6712 = 79, 7512 = 89, 8112 = 97, 8712 = 103, 9112 = 109, A712 = 127, 30112 = 433, 40112 = 577, 41512 = 593, 57512 = 809, 5BB12 = 863, 66112 = 937, 6A712 = 991

13

513 = 5, 4113 = 53, 6113 = 79, A113 = 131, C113 = 157, 40113 = 677

14

714 = 7, D14 = 13, 1514 = 19, 2114 = 29, 2314 = 31, 3114 = 43, 4B14 = 67, 5114 = 71, 5914 = 79, 6D14 = 97, 8114 = 113, 9114 = 127, 10114 = 197, 27114 = 491, 47114 = 883

15

515 = 5, B15 = 11, D15 = 13, 2115 = 31, 4115 = 61, 4B15 = 71, A115 = 151, A715 = 157, C115 = 181, CD15 = 193, D415 = 199, E115 = 211, 31B15 = 701

16

516 = 5, 716 = 7, D16 = 13, 1116 = 17, 6116 = 97, 7116 = 113, 7F16 = 127, C116 = 193, F116 = 241, 10116 = 257, 1C116 = 449, 24116 = 577, 28116 = 641, 30116 = 769

17

D17 = 13, 6117 = 103, 8117 = 137, E117 = 239

18

718 = 7, D18 = 13, H18 = 17, 1118 = 19, 2118 = 37, 4118 = 73, 6118 = 109, 7118 = 127, 9118 = 163, A118 = 181, B118 = 199, F118 = 271, H118 = 307, 13118 = 379, 16118 = 433, 19118 = 487, 1C118 = 541, 26118 = 757, 29118 = 811, 2F118 = 919

19

719 = 7, 6D19 = 127, A119 = 191, C119 = 229, G319 = 307

20

520 = 5, B20 = 11, D20 = 13, H20 = 17, J20 = 19, 1B20 = 31, 2120 = 41, 3120 = 61, 3B20 = 71, 5120 = 101, 7920 = 149, 7B20 = 151, 7H20 = 157, 9120 = 181, 9B20 = 191, 9D20 = 193, 9J20 = 199, C120 = 241, CH20 = 257, DB20 = 271, E120 = 281, ED20 = 293, FB20 = 311, H920 = 349, 10120 = 401, 1A120 = 601

 

Se nn termina con le stesse cifre di n, anche nnn termina con le stesse cifre.

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