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Esclusionari (numeri)

Rappresentazione dei numeri 

Clifford Pickover chiama “esclusionari” i numeri naturali tali che una loro potenza, detta ordine del numero, contenga solo cifre diverse da quelle del numero stesso.

 

Conviene limitarsi a numeri formate da cifre differenti, altrimenti vi sono varie famiglie infinite; per esempio, un numero formato da una sequenza di 3 ha un quadrato che non contiene alcun 3: 333332 = 1111088889; analogamente un numero formato da una sequenza di 6 seguiti da un 7 ha un quadrato che non contiene né 6 né 7: 666672 = 4444488889.

 

Per un dato ordine, sono finiti in qualsiasi base, semplicemente perché sono in numero finito i numeri formati da cifre differenti.

Fissata la base inoltre, ogni numero è, con ogni probabilità, esclusionario al massimo in un numero finito di ordini, perché all’aumentare del numero di cifre quelle che compongono il numero prima o poi sembra debbano comparire, ma non è stato dimostrato (v. anche due).

Viceversa ogni intero n maggiore di 1 è esclusionario di ordine k nelle basi non inferiori a nk, perché la potenza è rappresentata con una sola cifra, diversa da n.

 

Non esistono numeri esclusionari in base 2.

 

In nessuna base un numero esclusionario di qualsiasi ordine può terminare con 0 o 1; in base b pari nessun numero esclusionario di qualsiasi ordine può terminare con b / 2 + 1. Per esempio, nessun numero esclusionario in base 10 può terminare con 6.

 

La tabella seguente riporta i quadrati esclusionari (o numeri esclusionari del secondo ordine) inferiori a 1000 nelle basi sino a 20.

Base

Numeri esclusionari del secondo ordine

3

22 = 2

4

24 = 2, 34 = 3

5

25 = 2, 35 = 3, 45 = 4, 135 = 8

6

26 = 2, 56 = 5, 326 = 20, 356 = 23

7

27 = 2, 38 = 3, 48 = 4, 58 = 5, 68 = 6, 138 = 10, 148 = 11, 158 = 12, 268 = 20, 348 = 25, 528 = 37, 628 = 44, 1528 = 86, 2068 = 104, 3068 = 153, 3548 = 186, 5148 = 256, 6038 = 297, 16058 = 642

8

28 = 2, 38 = 3, 48 = 4, 58 = 5, 68 = 6, 78 = 7, 148 = 12, 168 = 14, 238 = 19, 258 = 21, 268 = 22, 528 = 42, 548 = 44, 568 = 46, 578 = 47, 628 = 50, 638 = 51, 648 = 52, 728 = 58, 1548 = 108, 1648 = 116, 1728 = 122, 1748 = 124, 2038 = 131, 2078 = 135, 2168 = 142, 2358 = 157, 2478 = 167, 2538 = 171, 2568 = 174, 2658 = 181, 3068= 198, 3078 = 199, 3478 = 231, 4638 = 307, 4658 = 309, 5068 = 326, 5148 = 332, 5268 = 342, 5628 = 370, 6038 = 387, 6058 = 389, 6238 = 403, 7038 = 451, 7148 = 460, 7328 = 474, 15748 = 892, 17268 = 982

9

29 = 2, 39 = 3, 49 = 4, 59 = 5, 69 = 6, 79 = 7, 89 = 8, 149 = 13, 159 = 14, 169 = 15, 239 = 21, 249 = 22, 259 = 23, 269 = 24, 349 = 31, 369 = 33, 379 = 34, 389 = 35, 439 = 39, 459 = 41, 469 = 42, 579 = 52, 589 = 53, 639 = 57, 679 = 61, 689 = 62, 829 = 74, 839 = 75, 1459 = 122, 1539 = 129, 1579 = 133, 1659 = 140, 1679 = 142, 1849 = 157, 2079 = 169, 2089 = 170, 2139 = 174, 2159 = 176, 2349 = 193, 2369 = 195, 2539 = 210, 2589 = 215, 2659 = 221, 2689 = 224, 3259 = 266, 3269 = 267, 3279 = 268, 3659 = 302, 3689 = 305, 3729 = 308, 3789 = 314, 4089 = 332, 4389 = 359, 4569 = 375, 5139 = 417, 5169 = 420, 5249 = 427, 5269 = 429, 5729 = 470, 6159 = 500, 6259 = 509, 6379 = 520, 6539 = 534, 6579 = 538, 6789 = 557, 6849 = 562, 7389 = 602, 8049 = 652, 8149 = 661, 8269 = 672

10

2, 3, 4, 7, 8, 9, 17, 18, 24, 29, 34, 38, 39, 47, 53, 54, 57, 58, 59, 62, 67, 72, 79, 84, 92, 94, 157, 158, 173, 187, 192, 194, 209, 237, 238, 247, 253, 257, 259, 307, 314, 349, 359, 409, 437, 459, 467, 547, 567, 612, 638, 659, 672, 673, 689, 712, 729, 738, 739, 749, 789, 807, 812, 813, 814, 834, 853, 854, 862, 904, 924

11

211 = 2, 311 = 3, 411 = 4, 511 = 5, 611 = 6, 711 = 7, 811 = 8, 911 = 9, A11 = 10, 1611 = 17, 1811 = 19, 1911 = 20, 2311 = 25, 2411 = 26, 2511 = 27, 2711 = 29, 2811 = 30, 2911 = 31, 2A11 = 32, 3211 = 35, 3711 = 40, 3811 = 41, 4311 = 47, 4511 = 49, 4611 = 50, 4711 = 51, 4811 = 52, 4A11 = 54, 5611 = 61, 6411 = 70, 6511 = 71, 6711 = 73, 6811 = 74, 6911 = 75, 6A11 = 76, 7311 = 80, 7611 = 83, 7911 = 86, 8211 = 90, 8311 = 91, 8411 = 92, 8A11 = 98, 9211 = 101, 9411 = 103, 9511 = 104, 9611 = 105, A211 = 112, A311 = 113, 14711 = 172, 14811 = 173, 15311 = 179, 15611 = 182, 17511 = 203, 17611 = 204, 18211 = 211, 18711 = 216, 19211 = 222, 1A211 = 233, 1A511 = 236, 1A611 = 237, 20911 = 251, 20A11 = 252, 21911 = 262, 23811 = 283, 23911 = 284, 24611 = 292, 24711 = 293, 24811 = 294, 25311 = 300, 28611 = 336, 28A11 = 340, 29511 = 346, 2A311 = 355, 2A511 = 357, 2A711 = 359, 30411 = 367, 30811 = 371, 30A11 = 373, 31811 = 382, 32711 = 392, 32811 = 393, 32911 = 394, 34A11 = 417, 36411 = 433, 36711 = 436, 36811 = 437, 36911 = 438, 36A11 = 439, 37411 = 444, 37911 = 449, 37A11 = 450, 39A11 = 472, 3A211 = 475, 3A711 = 480, 40811 = 492, 40A11 = 494, 42311 = 509, 43811 = 525, 45311 = 542, 45611 = 545, 46711 = 557, 46811 = 558, 47611 = 567, 48611= 578, 48711 = 579, 49611 = 589, 49A11 = 593, 50911 = 614, 50A11= 615, 51611 = 622, 54611 = 655, 54A11 = 659, 56A11 = 681, 57911 = 691, 58611 = 699, 59611 = 710, 61211 = 739, 61811 = 745, 64511 = 775, 67311 = 806, 67911 = 812, 68A11 = 824, 6A311 = 839, 6A511 = 841, 6A811 = 844, 6A911 = 845, 70811 = 855, 70A11 = 857, 71311 = 861, 72511 = 874, 72A11 = 879, 73211 = 882, 73811 = 888, 74811 = 899, 76811 = 921, 76A11 = 923, 78211 = 937, 78311 = 938, 78611= 941, 7A611 = 963, 7A911 = 966, 80411 = 972, 80711 = 975, 80911 = 977, 81311 = 982, 81411 = 983, 81911 = 988, 82411 = 994, 82511 = 995

12

212 = 2, 312 = 3, 512 = 5, 612 = 6, 712 = 7, 812 = 8, A12 = 10, B12 = 11, 1612 = 18, 1812 = 20, 1A12 = 22, 2312 = 27, 2712 = 31, 2812 = 32, 2A12 = 34, 2B12 = 35, 3212 = 38, 3512 = 41, 3712 = 43, 3812 = 44, 3A12 = 46, 4312 = 51, 4512 = 53, 4612 = 54, 4712 = 55, 4B12 = 59, 7212 = 86, 7312 = 87, 7612 = 90, 7812 = 92, 7A12 = 94, 7B12 = 95, 8A12 = 106, 8B12 = 107, 9212 = 110, 9512 = 113, 9A12 = 118, 9B12 = 119, A212 = 122, A312 = 123, A512 = 125, A612 = 126, A712 = 127, A812 = 128, B212 = 134, B312 = 135, B512 = 137, 15312 = 207, 15A12 = 214, 16812 = 224, 17312 = 231, 17A12 = 238, 18312 = 243, 18612 = 246, 18A12 = 250, 19A12 = 262, 1A212 = 266, 1A812 = 272, 1B212 = 278, 1B612 = 282, 20A12 = 298, 20B12 = 299, 21812 = 308, 21A12 = 310, 23A12 = 334, 24312 = 339, 24612 = 342, 24712 = 343, 25812 = 356, 27312 = 375, 27512 = 377, 27812 = 380, 28512 = 389, 28612 = 390, 28B12 = 395, 29812 = 404, 29A12 = 406, 2A312 = 411, 2A712 = 415, 2B312 = 423, 2B512 = 425, 2B612 = 426, 30512 = 437, 30B12 = 443, 31812 = 452, 32712 = 463, 32812 = 464, 35712 = 499, 36B12 = 515, 37512 = 521, 37812 = 524, 37A12 = 526, 37B12 = 527, 38A12 = 538, 38B12 = 539, 39712 = 547, 39A12 = 550, 39B12 = 551, 3A512 = 557, 42312 = 603, 42B12 = 611, 43712 = 619, 45B12 = 647, 47512 = 665, 47B12 = 671, 48512 = 677, 48612 = 678, 49512 = 689, 49712 = 691, 4A312 = 699, 4A612 = 702, 4AB12 = 707, 4B612 = 714, 50A12 = 730, 50B12 = 731, 51312 = 735, 51612 = 738, 51812 = 740, 51A12 = 742, 53712 = 763, 56A12 = 802, 56B12 = 803, 57B12 = 815, 64B12 = 923, 67A12 = 958, 67B12 = 959, 69512 = 977, 69B12 = 983, 6A712 = 991, 6A812 = 992, 6B212 = 998

13

213 = 2, 313 = 3, 413 = 4, 513 = 5, 613 = 6, 713 = 7, 813 = 8, 913 = 9, A13 = 10, B13 = 11, C13 = 12, 1713 = 20, 1813 = 21, 1913 = 22, 1B13 = 24, 2313 = 29, 2713 = 33, 2813 = 34, 2913 = 35, 2A13 = 36, 2B13 = 37, 2C13 = 38, 3213 = 41, 3613 = 45, 3713 = 46, 3813 = 47, 3A13 = 49, 3B13 = 50, 3C13 = 51, 4513 = 57, 4A13 = 62, 4C13 = 64, 5313 = 68, 5413 = 69, 5613 = 71, 5713 = 72, 5A13 = 75, 5B13 = 76, 5C13 = 77, 6313 = 81, 6413 = 82, 6513 = 83, 6713 = 85, 6813 = 86, 6913 = 87, 7513 = 96, 7613 = 97, 7813 = 99, 7A13 = 101, 7C13 = 103, 8213 = 106, 8313 = 107, 8913 = 113, 8A13 = 114, 8B13 = 115, 9213 = 119, 9413 = 121, 9513 = 122, 9713 = 124, 9813 = 125, 9B13 = 128, 9C13 = 129, A213 = 132, A313 = 133, A413 = 134, AB13 = 141, AC13 = 142, B213 = 145, B313 = 146, B513 = 148, B613 = 149, B713 = 150, C213 = 158, C313 = 159, C413 = 160, 15613 = 240, 16313 = 250, 16413 = 251, 16513 = 252, 16713 = 254, 16813 = 255, 16913 = 256, 17413 = 264, 17913 = 269, 18513 = 278, 18613 = 279, 18913 = 282, 18B13 = 284, 19413 = 290, 19513 = 291, 19713 = 293, 19813 = 294, 1A913 = 308, 1AB13 = 310, 1B513 = 317, 1B613 = 318, 1BA13 = 322, 1C213 = 327, 1C413 = 329, 1C613 = 331, 1C713 = 332, 20513 = 343, 20A13 = 348, 20B13 = 349, 20C13 = 350, 21313 = 354, 21713 = 358, 21913 = 360, 21B13 = 362, 23613 = 383, 24813 = 398, 24913 = 399, 24C13 = 402, 27A13 = 439, 27C13 = 441, 28313 = 445, 28413 = 446, 28513 = 447, 28613 = 448, 28913 = 451, 28A13 = 452, 29813 = 463, 29B13 = 466, 29C13 = 467, 2A313 = 471, 2A913 = 477, 2AB13 = 479, 2AC13 = 480, 2B513 = 486, 2B613 = 487, 2B713 = 488, 2BA13 = 491, 2C313 = 497, 2C413 = 498, 2C613 = 500, 2CA13 = 504, 30513 = 512, 30613 = 513, 30A13 = 517, 30B13 = 518, 30C13 = 519, 31213 = 522, 31713 = 527, 31813 = 528, 32513 = 538, 32B13 = 544, 34513 = 564, 35213 = 574, 35613 = 578, 35813 = 580, 36213 = 587, 36813 = 593, 37513 = 603, 37613 = 604, 37813 = 606, 37C13 = 610, 38713 = 618, 38A13 = 621, 39513 = 629, 39713 = 631, 39813 = 632, 39B13 = 635, 39C13 = 636, 3AB13 = 648, 3AC13 = 649, 3B713 = 657, 3BC13 = 662, 3C713 = 670, 3CA13 = 673, 40513 = 681, 40613 = 682, 40913 = 685, 40A13 = 686, 40C13 = 688, 43613 = 721, 43C13 = 727, 45A13 = 751, 45C13 = 753, 46313 = 757, 46513 = 759, 47913 = 776, 47C13 = 779, 4A313 = 809, 4B313 = 822, 4B513 = 824, 4C613 = 838, 4C713 = 839, 50413 = 849, 50613 = 851, 50813 = 853, 50913 = 854, 50A13 = 855, 50B13 = 856, 51413 = 862, 51813 = 866, 53713 = 891, 53813 = 892, 53B13 = 895, 54613 = 903, 56713 = 930, 58A13 = 959, 58B13 = 960, 59713 = 969, 59C13 = 974, 5A313 = 978, 5A413 = 979, 5AB13 = 986, 5B613 = 994, 5B713 = 995

14

214 = 2, 314 = 3, 414 = 4, 514 = 5, 614 = 6, 914 = 9, A14 = 10, B14 = 11, C14 = 12, D14 = 13, 1614 = 20, 1A14 = 24, 1B14 = 25, 1C14 = 26, 2314 = 31, 2614 = 34, 2914 = 37, 2B14 = 39, 2D14 = 41, 3214 = 44, 3414 = 46, 3614 = 48, 3A14 = 52, 3B14 = 53, 3D14 = 55, 4614 = 62, 4914 = 65, 4A14 = 66, 4B14 = 67, 5214 = 72, 5314 = 73, 5414 = 74, 5914 = 79, 5A14 = 80, 5C14 = 82, 5D14 = 83, 6314 = 87, 6414 = 88, 6914 = 93, 6A14 = 94, 6B14 = 95, 6C14 = 96, 7214 = 100, 7414 = 102, 7514 = 103, 7614 = 104, 7914 = 107, 7B14 = 109, 7C14 = 110, 7D14 = 111, 8214 = 114, 8314 = 115, 8514 = 117, 8A14 = 122, 8B14 = 123, 8D14 = 125, 9214 = 128, 9414 = 130, 9514 = 131, 9C14 = 138, 9D14 = 139, A214 = 142, A314 = 143, A414 = 144, A514 = 145, A914 = 149, AB14 = 151, AD14 = 153, B314 = 157, B414 = 158, B614 = 160, BC14 = 166, BD14 = 167, C214 = 170, C314 = 171, C414 = 172, C914 = 177, CA14 = 178, D214 = 184, D314 = 185, D414 = 186, D614 = 188, 15C14 = 278, 16B14 = 291, 17314 = 297, 17614 = 300, 17914 = 303, 17B14 = 305, 18414 = 312, 18514 = 313, 18C14 = 320, 19414 = 326, 19514 = 327, 19614 = 328, 19A14 = 332, 1A314 = 339, 1B414 = 354, 1B614 = 356, 1BA14 = 360, 1BC14 = 362, 1C214 = 366, 1C514 = 369, 1CA14 = 374, 1CB14 = 375, 1D214 = 380, 1D414 = 382, 1D614 = 384, 20514 = 397, 20C14 = 404, 20D14 = 405, 21314 = 409, 21614 = 412, 21C14 = 418, 23614 = 440, 24614 = 454, 24914 = 457, 24B14 = 459, 26314 = 479, 27314 = 493, 27914 = 499, 28314 = 507, 28C14 = 516, 29D14 = 531, 2A314 = 535, 2A514 = 537, 2AB14 = 543, 2B614 = 552, 2BC14 = 558, 2BD14 = 559, 2C514 = 565, 2C914 = 569, 2D314 = 577, 2D614 = 580, 2D914 = 583, 2DB14 = 585, 30414 = 592, 30514 = 593, 30614 = 594, 30A14 = 598, 30B14 = 599, 30C14 = 600, 30D14 = 601, 31214 = 604, 31414 = 606, 31614 = 608, 31B14 = 613, 31C14 = 614, 32514 = 621, 32614 = 622, 34514 = 649, 34614 = 650, 34914 = 653, 35214 = 660, 35914 = 667, 35A14 = 668, 35C14 = 670, 35D14 = 671, 36214 = 674, 36414 = 676, 36914 = 681, 36A14 = 682, 36B14 = 683, 37514 = 691, 37914 = 695, 37B14 = 697, 37D14 = 699, 38514 = 705, 38A14 = 710, 38C14 = 712, 39214 = 716, 39C14 = 726, 3AD14 = 741, 3B614 = 748, 3C214 = 758, 3C414 = 760, 3C514 = 761, 3CB14 = 767, 3CD14 = 769, 3D914 = 779, 3DB14 = 781, 40514 = 789, 40914 = 793, 40D14 = 797, 42614 = 818, 42914 = 821, 42D14 = 825, 47314 = 885, 47514 = 887, 47B14 = 893, 47D14 = 895, 48314 = 899, 48514 = 901, 48A14 = 906, 48D14 = 909, 49514 = 915, 49614 = 916, 49A14 = 920, 49D14 = 923, 4A514 = 929, 4B314 = 941, 4B614 = 944, 4BA14 = 948, 4C314 = 955, 4C514 = 957, 4C914 = 961, 4CB14 = 963, 4CD14 = 965, 4D314 = 969, 4D914 = 975, 50414 = 984, 50614 = 986, 50A14 = 990, 50C14 = 992, 50D14 = 993

15

215 = 2, 315 = 3, 415 = 4, 515 = 5, 715 = 7, 815 = 8, 915 = 9, B15 = 11, C15 = 12, D15 = 13, E15 = 14, 1715 = 22, 1815 = 23, 1915 = 24, 1C15 = 27, 1D15 = 28, 2315 = 33, 2715 = 37, 2815 = 38, 2915 = 39, 2B15 = 41, 2D15 = 43, 2E15 = 44, 3215 = 47, 3415 = 49, 3515 = 50, 3715 = 52, 3815 = 53, 3915 = 54, 3B15 = 56, 3C15 = 57, 3D15 = 58, 3E15 = 59, 4315 = 63, 4515 = 65, 4915 = 69, 4B15 = 71, 4C15 = 72, 4E15 = 74, 5315 = 78, 5415 = 79, 5715 = 82, 5815 = 83, 5B15 = 86, 5C15 = 87, 5D15 = 88, 6515 = 95, 6715 = 97, 6815 = 98, 6B15 = 101, 6C15 = 102, 6D15 = 103, 6E15 = 104, 7215 = 107, 7515 = 110, 7815 = 113, 7915 = 114, 7B15 = 116, 8515 = 125, 8715 = 127, 8915 = 129, 8B15 = 131, 8C15 = 132, 8D15 = 133, 8E15 = 134, 9215 = 137, 9415 = 139, 9815 = 143, 9D15 = 148, 9E15 = 149, A315 = 153, A415 = 154, A815 = 158, A915 = 159, AC15 = 162, B215 = 167, B315 = 168, B415 = 169, B515 = 170, B915 = 174, BC15 = 177, BD15 = 178, BE15 = 179, C415 = 184, C515 = 185, C715 = 187, C815 = 188, C915 = 189, CD15 = 193, CE15 = 194, D215 = 197, D415 = 199, D515 = 200, D715 = 202, D815 = 203, D915 = 204, DB15 = 206, E215 = 212, E315 = 213, E415 = 214, E515 = 215, 16715 = 322, 16815 = 323, 17315 = 333, 17515 = 335, 17815 = 338, 17915 = 339, 18515 = 350, 18715 = 352, 18C15 = 357, 18D15 = 358, 19D15 = 373, 1A315 = 378, 1A715 = 382, 1A815 = 383, 1AC15 = 387, 1B215 = 392, 1B915 = 399, 1BC15 = 402, 1C715 = 412, 1C915 = 414, 1CD15 = 418, 1D215 = 422, 1D515 = 425, 1D715 = 427, 1D815 = 428, 1DC15 = 432, 1E215 = 437, 1E715 = 442, 1E815 = 443, 1E915 = 444, 20715 = 457, 20C15 = 462, 20D15 = 463, 20E15 = 464, 21315 = 468, 21815 = 473, 21915 = 474, 21C15 = 477, 21D15 = 478, 23515 = 500, 23715 = 502, 23815 = 503, 23B15 = 506, 23D15 = 508, 23E15 = 509, 24B15 = 521, 24C15 = 522, 24E15 = 524, 26715 = 547, 26815 = 548, 26B15 = 551, 27815 = 563, 27915 = 564, 27B15 = 566, 28415 = 574, 28515 = 575, 28B15 = 581, 28C15 = 582, 28D15 = 583, 28E15 = 584, 29415 = 589, 29D15 = 598, 29E15 = 599, 2A415 = 604, 2A815 = 608, 2A915 = 609, 2AB15 = 611, 2AC15 = 612, 2AD15 = 613, 2B415 = 619, 2B515 = 620, 2B815 = 623, 2B915 = 624, 2BE15 = 629, 2C415 = 634, 2C515 = 635, 2C715 = 637, 2C915 = 639, 2D715 = 652, 2D915 = 654, 2DB15 = 656, 2E415 = 664, 2E515 = 665, 2E715 = 667, 2EB15 = 671, 30415 = 679, 30B15 = 686, 30E15 = 689, 31215 = 692, 31715 = 697, 31815 = 698, 31D15 = 703, 32815 = 713, 32915 = 714, 32B15 = 716, 34915 = 744, 34B15 = 746, 34C15 = 747, 35715 = 757, 35C15 = 762, 35D15 = 763, 36215 = 767, 36515 = 770, 36715 = 772, 36815 = 773, 36D15 = 778, 37415 = 784, 37515 = 785, 37915 = 789, 37B15 = 791, 38415 = 799, 38515 = 800, 38915 = 804, 38B15 = 806, 38E15 = 809, 39215 = 812, 39515 = 815, 39E15 = 824, 3A415 = 829, 3A915 = 834, 3AE15 = 839, 3B215 = 842, 3B415 = 844, 3B515 = 845, 3BD15 = 853, 3C415 = 859, 3C515 = 860, 3C715 = 862, 3C815 = 863, 3C915 = 864, 3D215 = 872, 3D415 = 874, 3D515 = 875, 3DB15 = 881, 3DC15 = 882, 3DE15 = 884, 3E215 = 887, 3E415 = 889, 3E515 = 890, 3E715 = 892, 3E915 = 894, 3EB15 = 896, 3EC15 = 897, 40515 = 905, 40B15 = 911, 40C15 = 912, 40E15 = 914, 43515 = 950

16

216 = 2, 316 = 3, 416 = 4, 516 = 5, 616 = 6, 716 = 7, 816 = 8, 916 = 9, A16 = 10, B16 = 11, C16 = 12, D16 = 13, E16 = 14, F16 = 15, 1816 = 24, 1B16 = 27, 1D16 = 29, 1E16 = 30, 2316 = 35, 2416 = 36, 2616 = 38, 2716 = 39, 2816 = 40, 2A16 = 42, 2B16 = 43, 2C16 = 44, 2D16 = 45, 2E16 = 46, 2F16 = 47, 3216 = 50, 3416 = 52, 3516 = 53, 3716 = 55, 3816 = 56, 3916 = 57, 3A16 = 58, 3B16 = 59, 3C16 = 60, 3D16 = 61, 3E16 = 62, 4316 = 67, 4516 = 69, 4716 = 71, 4B16 = 75, 4C16 = 76, 4D16 = 77, 4F16 = 79, 5216 = 82, 5316 = 83, 5416 = 84, 5616 = 86, 5716 = 87, 5816 = 88, 5916 = 89, 5C16 = 92, 5D16 = 93, 5E16 = 94, 5F16 = 95, 6416 = 100, 6516 = 101, 6816 = 104, 6916 = 105, 6A16 = 106, 6C16 = 108, 6E16 = 110, 6F16 = 111, 9216 = 146, 9616 = 150, 9816 = 152, 9C16 = 156, 9E16 = 158, 9F16 = 159, A216 = 162, A316 = 163, A416 = 164, A716 = 167, A816 = 168, AB16 = 171, AC16 = 172, AD16 = 173, AE16 = 174, B316 = 179, B416 = 180, B516 = 181, B716 = 183, BA16 = 186, BC16 = 188, BE16 = 190, BF16 = 191, C216 = 194, C316 = 195, C416 = 196, C516 = 197, C616 = 198, C716 = 199, CA16 = 202, CB16 = 203, CD16 = 205, CF16 = 207, D216 = 210, D416 = 212, D516 = 213, D616 = 214, D716 = 215, D816 = 216, D916 = 217, DE16 = 222, DF16 = 223, E216 = 226, E316 = 227, E416 = 228, E516 = 229, E816 = 232, E916 = 233, EC16 = 236, F216 = 242, F316 = 243, F416 = 244, F516 = 245, F716 = 247, 16C16 = 364, 16E16 = 366, 17A16 = 378, 17E16 = 382, 18316 = 387, 18616 = 390, 18A16 = 394, 18C16 = 396, 18D16 = 397, 18E16 = 398, 19616 = 406, 1A816 = 424, 1AB16 = 427, 1AD16 = 429, 1AE16 = 430, 1B416 = 436, 1B516 = 437, 1B816 = 440, 1BD16 = 445, 1BE16 = 446, 1C616 = 454, 1CA16 = 458, 1CB16 = 459, 1CD16 = 461, 1D216 = 466, 1D416 = 468, 1D616 = 470, 1D816 = 472, 1DE16 = 478, 1E216 = 482, 1E516 = 485, 1E616 = 486, 1E816 = 488, 1ED16 = 493, 1F216 = 498, 1F416 = 500, 1F516 = 501, 1F816 = 504, 1FA16 = 506, 20516 = 517, 20716 = 519, 20D16 = 525, 20E16 = 526, 20F16 = 527, 21316 = 531, 21616 = 534, 21B16 = 539, 21D16 = 541, 21E16 = 542, 23516 = 565, 23816 = 568, 23916 = 569, 23B16 = 571, 23C16 = 572, 23E16 = 574, 23F16 = 575, 24316 = 579, 24916 = 585, 26416 = 612, 26816 = 616, 26916 = 617, 26A16 = 618, 26C16 = 620, 27316 = 627, 27916 = 633, 27D16 = 637, 27E16 = 638, 27F16 = 639, 28316 = 643, 28716 = 647, 28916 = 649, 28B16 = 651, 29716 = 663, 29816 = 664, 29F16 = 671, 2A416 = 676, 2A816 = 680, 2AB16 = 683, 2B816 = 696, 2BC16 = 700, 2BD16 = 701, 2BE16 = 702, 2BF16 = 703, 2C316 = 707, 2C416 = 708, 2C516 = 709, 2CF16 = 719, 2D316 = 723, 2D416 = 724, 2D616 = 726, 2D716 = 727, 2D916 = 729, 2DE16 = 734, 2DF16 = 735, 2E316 = 739, 2E416 = 740, 2E516 = 741, 2E916 = 745, 2EC16 = 748, 2ED16 = 749, 2F416 = 756, 2F516 = 757, 2F716 = 759, 2F916 = 761, 2FC16 = 764, 2FD16 = 765, 30516 = 773, 30616 = 774, 30B16 = 779, 30D16 = 781, 30E16 = 782, 30F16 = 783, 31216 = 786, 31416 = 788, 31516 = 789, 31616 = 790, 31816 = 792, 31B16 = 795, 31E16 = 798, 32416 = 804, 32616 = 806, 32716 = 807, 32816 = 808, 32C16 = 812, 32D16 = 813, 32E16 = 814, 34516 = 837, 34716 = 839, 34916 = 841, 34B16 = 843, 34C16 = 844, 34D16 = 845, 35216 = 850, 35616 = 854, 35816 = 856, 35C16 = 860, 35D16 = 861, 36216 = 866, 36416 = 868, 36816 = 872, 36916 = 873, 36C16 = 876, 36F16 = 879, 37216 = 882, 37416 = 884, 37A16 = 890, 37B16 = 891, 37D16 = 893, 38216 = 898, 38416 = 900, 38916 = 905, 38D16 = 909, 38E16 = 910, 38F16 = 911, 39216 = 914, 39616 = 918, 39816 = 920, 39C16 = 924, 39E16 = 926, 39F16 = 927, 3A716 = 935, 3A816 = 936, 3AC16 = 940, 3B216 = 946, 3B716 = 951, 3B816 = 952, 3BC16 = 956, 3BF16 = 959, 3C216 = 962, 3C416 = 964, 3C716 = 967, 3CA16 = 970, 3CB16 = 971, 3CD16 = 973, 3CF16 = 975, 3D216 = 978, 3D416 = 980, 3D716 = 983, 3D816 = 984, 3DC16 = 988, 3E516 = 997

17

217 = 2, 317 = 3, 417 = 4, 517 = 5, 617 = 6, 717 = 7, 817 = 8, 917 = 9, A17 = 10, B17 = 11, C17 = 12, D17 = 13, E17 = 14, F17 = 15, G17 = 16, 1817 = 25, 1917 = 26, 1A17 = 27, 1B17 = 28, 1C17 = 29, 1E17 = 31, 1F17 = 32, 2317 = 37, 2717 = 41, 2817 = 42, 2917 = 43, 2A17 = 44, 2C17 = 46, 2D17 = 47, 2E17 = 48, 2F17 = 49, 2G17 = 50, 3217 = 53, 3417 = 55, 3517 = 56, 3617 = 57, 3717 = 58, 3817 = 59, 3917 = 60, 3A17 = 61, 3B17 = 62, 3D17 = 64, 3G17 = 67, 4317 = 71, 4517 = 73, 4617 = 74, 4817 = 76, 4917 = 77, 4C17 = 80, 4D17 = 81, 4G17 = 84, 5217 = 87, 5317 = 88, 5417 = 89, 5617 = 91, 5717 = 92, 5817 = 93, 5A17 = 95, 5B17 = 96, 5C17 = 97, 5D17 = 98, 5E17 = 99, 5F17 = 100, 6317 = 105, 6817 = 110, 6917 = 111, 6B17 = 113, 6C17 = 114, 6D17 = 115, 6E17 = 116, 6G17 = 118, 7417 = 123, 7517 = 124, 7617 = 125, 7817 = 127, 7917 = 128, 7A17 = 129, 7C17 = 131, 7D17 = 132, 7E17 = 133, 7F17 = 134, 7G17 = 135, 8217 = 138, 8417 = 140, 8617 = 142, 8717 = 143, 8917 = 145, 8A17 = 146, 8B17 = 147, 9617 = 159, 9817 = 161, 9A17 = 163, 9B17 = 164, 9D17 = 166, 9F17 = 168, 9G17 = 169, A217 = 172, A317 = 173, A417 = 174, A517 = 175, A717 = 177, A917 = 179, AE17 = 184, AF17 = 185, AG17 = 186, B217 = 189, B317 = 190, B417 = 191, B517 = 192, B617 = 193, B817 = 195, B917 = 196, BA17 = 197, BC17 = 199, BD17 = 200, BE17 = 201, BF17 = 202, BG17 = 203, C217 = 206, C417 = 208, C517 = 209, C617 = 210, C717 = 211, CA17 = 214, CB17 = 215, CD17 = 217, CE17 = 218, CG17 = 220, D417 = 225, D517 = 226, D617 = 227, D717 = 228, DB17 = 232, DC17 = 233, DE17 = 235, DF17 = 236, DG17 = 237, E217 = 240, E317 = 241, E417 = 242, E617 = 244, E717 = 245, E817 = 246, E917 = 247, EA17 = 248, EB17 = 249, EF17 = 253, EG17 = 254, F217 = 257, F317 = 258, F417 = 259, F617 = 261, F817 = 263, F917 = 264, FB17 = 266, FC17 = 267, FD17 = 268, G217 = 274, G317 = 275, G517 = 277, G617 = 278, G717 = 279, 17317 = 411, 17417 = 412, 17517 = 413, 17E17 = 422, 18417 = 429, 18617 = 431, 18717 = 432, 18917 = 434, 19517 = 447, 19817 = 450, 19C17 = 454, 19D17 = 455, 19F17 = 457, 1A317 = 462, 1A517 = 464, 1AD17 = 472, 1AE17 = 473, 1AF17 = 474, 1B417 = 480, 1B617 = 482, 1B717 = 483, 1B917 = 485, 1BC17 = 488, 1C517 = 498, 1C617 = 499, 1C717 = 500, 1C917 = 502, 1CA17 = 503, 1D517 = 515, 1D617 = 516, 1D717 = 517, 1DB17 = 521, 1DC17 = 522, 1E217 = 529, 1E417 = 531, 1E817 = 535, 1EA17 = 537, 1EB17 = 538, 1EC17 = 539, 1ED17 = 540, 1EF17 = 542, 1F217 = 546, 1F617 = 550, 1F817 = 552, 1F917 = 553, 1FD17 = 557, 1FE17 = 558, 1G217 = 563, 1G517 = 566, 1G617 = 567, 1G717 = 568, 1G817 = 569, 1G917 = 570, 1GA17 = 571, 1GB17 = 572, 20517 = 583, 20817 = 586, 20D17 = 591, 20E17 = 592, 20F17 = 593, 20G17 = 594, 21317 = 598, 21717 = 602, 21917 = 604, 21A17 = 605, 21C17 = 607, 21E17 = 609, 21F17 = 610, 23517 = 634, 23717 = 636, 23817 = 637, 23917 = 638, 23A17 = 639, 23F17 = 644, 24317 = 649, 24D17 = 659, 26417 = 684, 26717 = 687, 26817 = 688, 26917 = 689, 26D17 = 693, 26E17 = 694, 26G17 = 696, 27317 = 700, 27417 = 701, 27A17 = 707, 27C17 = 709, 27D17 = 710, 27E17 = 711, 27G17 = 713, 28717 = 721, 28917 = 723, 28A17 = 724, 29517 = 736, 29C17 = 743, 29D17 = 744, 29F17 = 746, 2A317 = 751, 2A417 = 752, 2A517 = 753, 2A717 = 755, 2AD17 = 761, 2AF17 = 763, 2B317 = 768, 2B417 = 769, 2BA17 = 775, 2BC17 = 777, 2BD17 = 778, 2BE17 = 779, 2BF17 = 780, 2C317 = 785, 2C917 = 791, 2CA17 = 792, 2CD17 = 795, 2CE17 = 796, 2CF17 = 797, 2CG17 = 798, 2D317 = 802, 2D517 = 804, 2DC17 = 811, 2DE17 = 813, 2E317 = 819, 2EC17 = 828, 2EF17 = 831, 2EG17 = 832, 2F317 = 836, 2F417 = 837, 2FC17 = 845, 2FD17 = 846, 2FE17 = 847, 2G317 = 853, 2G517 = 855, 2G717 = 857, 2G917 = 859, 2GC17 = 862, 30517 = 872, 30617 = 873, 30717 = 874, 30C17 = 879, 30D17 = 880, 30E17 = 881, 30F17 = 882, 30G17 = 883, 31217 = 886, 31417 = 888, 31517 = 889, 31717 = 891, 31A17 = 894, 31B17 = 895, 31C17 = 896, 31E17 = 898, 31F17 = 899, 32417 = 905, 32517 = 906, 32717 = 908, 32A17 = 911, 32C17 = 913, 32D17 = 914, 32E17 = 915, 32F17 = 916, 34517 = 940, 34817 = 943, 34917 = 944, 34B17 = 946, 34D17 = 948, 34G17 = 951, 35417 = 956, 35617 = 958, 35817 = 960, 35C17 = 964, 35E17 = 966, 36217 = 971, 36417 = 973, 36817 = 977, 36917 = 978, 36C17 = 981, 36D17 = 982, 36E17 = 983, 37217 = 988, 37417 = 990, 37617 = 992, 37817 = 994, 37917 = 995, 37A17 = 996, 37D17 = 999

18

218 = 2, 318 = 3, 418 = 4, 518 = 5, 618 = 6, 718 = 7, 818 = 8, B18 = 11, C18 = 12, D18 = 13, E18 = 14, F18 = 15, G18 = 16, H18 = 17, 1B18 = 29, 1C18 = 30, 1D18 = 31, 1E18 = 32, 1F18 = 33, 1G18 = 34, 2318 = 39, 2618 = 42, 2718 = 43, 2818 = 44, 2B18 = 47, 2D18 = 49, 2E18 = 50, 2F18 = 51, 2G18 = 52, 2H18 = 53, 3218 = 56, 3418 = 58, 3518 = 59, 3618 = 60, 3718 = 61, 3818 = 62, 3B18 = 65, 3C18 = 66, 3F18 = 69, 3G18 = 70, 3H18 = 71, 4318 = 75, 4618 = 78, 4818 = 80, 4C18 = 84, 4F18 = 87, 4H18 = 89, 5318 = 93, 5618 = 96, 5718 = 97, 5818 = 98, 5B18 = 101, 5C18 = 102, 5E18 = 104, 5F18 = 105, 6318 = 111, 6418 = 112, 6518 = 113, 6718 = 115, 6818 = 116, 6B18 = 119, 6C18 = 120, 6D18 = 121, 6E18 = 122, 6F18 = 123, 6G18 = 124, 6H18 = 125, 7318 = 129, 7418 = 130, 7618 = 132, 7B18 = 137, 7C18 = 138, 7E18 = 140, 7G18 = 142, 7H18 = 143, 8218 = 146, 8418 = 148, 8518 = 149, 8718 = 151, 8B18 = 155, 8C18 = 156, 8D18 = 157, 8E18 = 158, 8F18 = 159, 8G18 = 160, 9218 = 164, 9518 = 167, 9618 = 168, 9718 = 169, 9818 = 170, 9B18 = 173, 9C18 = 174, 9E18 = 176, 9G18 = 178, 9H18 = 179, A218 = 182, A318 = 183, A418 = 184, A618 = 186, A718 = 187, AC18 = 192, AD18 = 193, AE18 = 194, AF18 = 195, AH18 = 197, B218 = 200, B318 = 201, B418 = 202, B518 = 203, B618 = 204, B818 = 206, BC18 = 210, BE18 = 212, BF18 = 213, BG18 = 214, BH18 = 215, C318 = 219, C418 = 220, C518 = 221, C618 = 222, C718 = 223, CB18 = 227, CD18 = 229, CE18 = 230, CG18 = 232, CH18 = 233, D218 = 236, D318 = 237, D418 = 238, D618 = 240, DC18 = 246, DF18 = 249, DH18 = 251, E318 = 255, E618 = 258, E718 = 259, E818 = 260, EC18 = 264, EF18 = 267, EG18 = 268, EH18 = 269, F218 = 272, F318 = 273, F518 = 275, F618 = 276, F718 = 277, F818 = 278, FB18 = 281, FC18 = 282, FG18 = 286, FH18 = 287, G218 = 290, G318 = 291, G518 = 293, G618 = 294, G818 = 296, GC18 = 300, GD18 = 301, H218 = 308, H318 = 309, H418 = 310, H518 = 311, H618 = 312, H718 = 313, H818 = 314, 17C18 = 462, 17E18 = 464, 18318 = 471, 18418 = 472, 18518 = 473, 18718 = 475, 18B18 = 479, 18C18 = 480, 18E18 = 482, 18F18 = 483, 18G18 = 484, 19418 = 490, 19718 = 493, 19818 = 494, 19B18 = 497, 19E18 = 500, 19G18 = 502, 1A518 = 509, 1A718 = 511, 1AC18 = 516, 1AD18 = 517, 1B818 = 530, 1BE18 = 536, 1BF18 = 537, 1C318 = 543, 1C418 = 544, 1C518 = 545, 1C618 = 546, 1C718 = 547, 1CD18 = 553, 1CE18 = 554, 1CG18 = 556, 1D318 = 561, 1D418 = 562, 1D618 = 564, 1DF18 = 573, 1E518 = 581, 1E718 = 583, 1EC18 = 588, 1EF18 = 591, 1F218 = 596, 1F418 = 598, 1F518 = 599, 1F618 = 600, 1FB18 = 605, 1FC18 = 606, 1FD18 = 607, 1FE18 = 608, 1FG18 = 610, 1G218 = 614, 1G518 = 617, 1G618 = 618, 1G718 = 619, 1GD18 = 625, 1GF18 = 627, 1H218 = 632, 1H418 = 634, 1H518 = 635, 1H718 = 637, 1H818 = 638, 1HB18 = 641, 20818 = 656, 20F18 = 663, 20G18 = 664, 20H18 = 665, 21318 = 669, 21618 = 672, 21718 = 673, 21B18 = 677, 21C18 = 678, 21D18 = 679, 21F18 = 681, 21G18 = 682, 23518 = 707, 23818 = 710, 23C18 = 714, 23D18 = 715, 23E18 = 716, 23H18 = 719, 24818 = 728, 24C18 = 732, 26318 = 759, 26418 = 760, 26718 = 763, 26818 = 764, 26B18 = 767, 26D18 = 769, 26E18 = 770, 26F18 = 771, 26G18 = 772, 26H18 = 773, 27318 = 777, 27618 = 780, 27C18 = 786, 27E18 = 788, 27F18 = 789, 27G18 = 790, 28318 = 795, 28418 = 796, 28518 = 797, 28E18 = 806, 28F18 = 807, 28G18 = 808, 28H18 = 809, 29418 = 814, 29818 = 818, 29B18 = 821, 29C18 = 822, 29E18 = 824, 29G18 = 826, 2A718 = 835, 2AH18 = 845, 2B318 = 849, 2B418 = 850, 2B518 = 851, 2B718 = 853, 2B818 = 854, 2BD18 = 859, 2BG18 = 862, 2BH18 = 863, 2C418 = 868, 2C518 = 869, 2C818 = 872, 2CB18 = 875, 2CD18 = 877, 2CE18 = 878, 2CH18 = 881, 2D318 = 885, 2D418 = 886, 2D618 = 888, 2DC18 = 894, 2DF18 = 897, 2DH18 = 899, 2E818 = 908, 2EB18 = 911, 2EF18 = 915, 2EG18 = 916, 2EH18 = 917, 2F318 = 921, 2F418 = 922, 2FB18 = 929, 2FC18 = 930, 2FD18 = 931, 2FE18 = 932, 2FG18 = 934, 2FH18 = 935, 2G518 = 941, 2G718 = 943, 2GC18 = 948, 2GD18 = 949, 2GF18 = 951, 2H318 = 957, 2H418 = 958, 2H518 = 959, 2H618 = 960, 2H718 = 961, 2HD18 = 967, 2HE18 = 968, 30518 = 977, 30718 = 979, 30D18 = 985, 30E18 = 986, 30G18 = 988, 30H18 = 989, 31218 = 992, 31418 = 994, 31518 = 995

19

219 = 2, 319 = 3, 419 = 4, 519 = 5, 619 = 6, 719 = 7, 819 = 8, 919 = 9, A19 = 10, B19 = 11, C19 = 12, D19 = 13, E19 = 14, F19 = 15, G19 = 16, H19 = 17, I19 = 18, 1819 = 27, 1919 = 28, 1A19 = 29, 1B19 = 30, 1D19 = 32, 1E19 = 33, 1F19 = 34, 1G19 = 35, 1H19 = 36, 2319 = 41, 2719 = 45, 2819 = 46, 2A19 = 48, 2B19 = 49, 2C19 = 50, 2D19 = 51, 2E19 = 52, 2F19 = 53, 2G19 = 54, 2H19 = 55, 2I19 = 56, 3219 = 59, 3419 = 61, 3519 = 62, 3619 = 63, 3719 = 64, 3819 = 65, 3919 = 66, 3A19 = 67, 3B19 = 68, 3E19 = 71, 3F19 = 72, 3G19 = 73, 3I19 = 75, 4319 = 79, 4519 = 81, 4619 = 82, 4719 = 83, 4819 = 84, 4919 = 85, 4A19 = 86, 4B19 = 87, 4C19 = 88, 4D19 = 89, 4E19 = 90, 4F19 = 91, 4I19 = 94, 5219 = 97, 5319 = 98, 5419 = 99, 5619 = 101, 5719 = 102, 5819 = 103, 5B19 = 106, 5E19 = 109, 5F19 = 110, 5G19 = 111, 5H19 = 112, 5I19 = 113, 6219 = 116, 6319 = 117, 6419 = 118, 6719 = 121, 6819 = 122, 6919 = 123, 6A19 = 124, 6B19 = 125, 6C19 = 126, 6F19 = 129, 6G19 = 130, 6H19 = 131, 6I19 = 132, 7319 = 136, 7419 = 137, 7519 = 138, 7619 = 139, 7919 = 142, 7A19 = 143, 7C19 = 145, 7D19 = 146, 7E19 = 147, 7F19 = 148, 7G19 = 149, 7H19 = 150, 7I19 = 151, 8419 = 156, 8619 = 158, 8719 = 159, 8919 = 161, 8A19 = 162, 8C19 = 164, 8D19 = 165, 8E19 = 166, 8F19 = 167, 8G19 = 168, 8H19 = 169, 8I19 = 170, 9219 = 173, 9619 = 177, 9719 = 178, 9819 = 179, 9A19 = 181, 9B19 = 182, 9C19 = 183, 9D19 = 184, 9E19 = 185, A619 = 196, A719 = 197, A819 = 198, A919 = 199, AB19 = 201, AC19 = 202, AD19 = 203, AE19 = 204, AF19 = 205, AH19 = 207, AI19 = 208, B219 = 211, B319 = 212, B519 = 214, B919 = 218, BA19 = 219, BD19 = 222, BF19 = 224, BG19 = 225, BH19 = 226, BI19 = 227, C219 = 230, C319 = 231, C419 = 232, C519 = 233, C719 = 235, C919 = 237, CA19 = 238, CB19 = 239, CD19 = 241, CE19 = 242, CF19 = 243, CH19 = 245, D219 = 249, D319 = 250, D419 = 251, D519 = 252, D819 = 255, DA19 = 257, DC19 = 259, DF19 = 262, DG19 = 263, DH19 = 264, DI19 = 265, E219 = 268, E319 = 269, E419 = 270, E519 = 271, E719 = 273, E819 = 274, EA19 = 276, EC19 = 278, ED19 = 279, EF19 = 281, EG19 = 282, EH19 = 283, F219 = 287, F319 = 288, F419 = 289, F519 = 290, F719 = 292, F919 = 294, FA19 = 295, FB19 = 296, FD19 = 298, FE19 = 299, FG19 = 301, FH19 = 302, FI19 = 303, G219 = 306, G319 = 307, G519 = 309, G619 = 310, G719 = 311, G819 = 312, G919 = 313, GA19 = 314, GC19 = 316, GD19 = 317, GH19 = 321, GI19 = 322, H219 = 325, H319 = 326, H419 = 327, H519 = 328, H719 = 330, H819 = 331, H919 = 332, HA19 = 333, HB19 = 334, HC19 = 335, HE19 = 337, HF19 = 338, I219 = 344, I319 = 345, I419 = 346, I519 = 347, I619 = 348, I819 = 350, 18319 = 516, 18419 = 517, 18C19 = 525, 18D19 = 526, 18E19 = 527, 18F19 = 528, 18G19 = 529, 18H19 = 530, 19419 = 536, 19519 = 537, 19619 = 538, 19719 = 539, 19A19 = 542, 19B19 = 543, 19C19 = 544, 19D19 = 545, 19E19 = 546, 1A619 = 557, 1A919 = 560, 1AB19 = 562, 1AC19 = 563, 1AD19 = 564, 1AF19 = 566, 1AH19 = 568, 1B319 = 573, 1B419 = 574, 1B519 = 575, 1B619 = 576, 1B919 = 579, 1BH19 = 587, 1C519 = 594, 1C719 = 596, 1C819 = 597, 1C919 = 598, 1CA19 = 599, 1CB19 = 600, 1CD19 = 602, 1CE19 = 603, 1CG19 = 605, 1CH19 = 606, 1D319 = 611, 1D419 = 612, 1D519 = 613, 1D619 = 614, 1D819 = 616, 1DA19 = 618, 1DB19 = 619, 1DC19 = 620, 1DF19 = 623, 1DG19 = 624, 1E219 = 629, 1EA19 = 637, 1EB19 = 638, 1EG19 = 643, 1EH19 = 644, 1F219 = 648, 1F519 = 651, 1F619 = 652, 1F719 = 653, 1F919 = 655, 1FB19 = 657, 1FC19 = 658, 1FD19 = 659, 1FE19 = 660, 1G519 = 670, 1G619 = 671, 1G719 = 672, 1G819 = 673, 1GA19 = 675, 1GC19 = 677, 1GD19 = 678, 1GE19 = 679, 1GH19 = 682, 1H219 = 686, 1H419 = 688, 1H519 = 689, 1H719 = 691, 1H819 = 692, 1H919 = 693, 1HA19 = 694, 1HB19 = 695, 1HC19 = 696, 1HF19 = 699, 1HG19 = 700, 1I219 = 705, 1I419 = 707, 1I619 = 709, 1I819 = 711, 1I919 = 712, 1IA19 = 713, 1IB19 = 714, 1IC19 = 715, 20519 = 727, 20619 = 728, 20819 = 730, 20919 = 731, 20F19 = 737, 20G19 = 738, 20H19 = 739, 20I19 = 740, 21319 = 744, 21719 = 748, 21819 = 749, 21919 = 750, 21A19 = 751, 21B19 = 752, 21D19 = 754, 21F19 = 756, 21G19 = 757, 21H19 = 758, 23519 = 784, 23619 = 785, 23719 = 786, 23819 = 787, 23919 = 788, 23A19 = 789, 23B19 = 790, 23D19 = 792, 23E19 = 793, 23I19 = 797, 24A19 = 808, 24B19 = 809, 24D19 = 811, 24F19 = 813, 24G19 = 814, 26319 = 839, 26419 = 840, 26719 = 843, 26C19 = 848, 26D19 = 849, 26F19 = 851, 26G19 = 852, 26H19 = 853, 27319 = 858, 27419 = 859, 27619 = 861, 27919 = 864, 27A19 = 865, 27C19 = 867, 27D19 = 868, 27E19 = 869, 27G19 = 871, 27H19 = 872, 28419 = 878, 28719 = 881, 28919 = 883, 28A19 = 884, 28E19 = 888, 28F19 = 889, 28G19 = 890, 29419 = 897, 29519 = 898, 29719 = 900, 29819 = 901, 29A19 = 903, 29C19 = 905, 29D19 = 906, 29E19 = 907, 2A519 = 917, 2A819 = 920, 2AE19 = 926, 2AF19 = 927, 2AH19 = 929, 2AI19 = 930, 2B419 = 935, 2B519 = 936, 2B919 = 940, 2BA19 = 941, 2BD19 = 944, 2BF19 = 946, 2BI19 = 949, 2C319 = 953, 2C419 = 954, 2C519 = 955, 2C819 = 958, 2C919 = 959, 2CA19 = 960, 2CG19 = 966, 2CH19 = 967, 2D419 = 973, 2D619 = 975, 2D919 = 978, 2DB19 = 980, 2DC19 = 981, 2DG19 = 985, 2DH19 = 986, 2DI19 = 987, 2E419 = 992, 2E619 = 994, 2E819 = 996, 2EA19 = 998, 2EB19 = 999

20

220 = 2, 320 = 3, 420 = 4, 620 = 6, 720 = 7, 820 = 8, 920 = 9, A20 = 10, B20 = 11, C20 = 12, D20 = 13, E20 = 14, F20 = 15, H20 = 17, I20 = 18, J20 = 19, 1A20 = 30, 1C20 = 32, 1D20 = 33, 1E20 = 34, 1H20 = 37, 1I20 = 38, 2320 = 43, 2620 = 46, 2720 = 47, 2820 = 48, 2920 = 49, 2A20 = 50, 2B20 = 51, 2C20 = 52, 2D20 = 53, 2E20 = 54, 2F20 = 55, 2I20 = 58, 2J20 = 59, 3220 = 62, 3620 = 66, 3720 = 67, 3820 = 68, 3920 = 69, 3A20 = 70, 3B20 = 71, 3E20 = 74, 3F20 = 75, 3H20 = 77, 3I20 = 78, 3J20 = 79, 4620 = 86, 4720 = 87, 4920 = 89, 4A20 = 90, 4B20 = 91, 4D20 = 93, 4E20 = 94, 4F20 = 95, 4H20 = 97, 5220 = 102, 5320 = 103, 5420 = 104, 5620 = 106, 5720 = 107, 5820 = 108, 5B20 = 111, 5C20 = 112, 5D20 = 113, 5E20 = 114, 5H20 = 117, 5I20 = 118, 5J20 = 119, 6220 = 122, 6320 = 123, 6420 = 124, 6820 = 128, 6920 = 129, 6A20 = 130, 6B20 = 131, 6C20 = 132, 6D20 = 133, 6E20 = 134, 6F20 = 135, 6I20 = 138, 7320 = 143, 7420 = 144, 7620 = 146, 7820 = 148, 7920 = 149, 7A20 = 150, 7B20 = 151, 7C20 = 152, 7D20 = 153, 7E20 = 154, 7F20 = 155, 7H20 = 157, 7I20 = 158, 7J20 = 159, 8220 = 162, 8720 = 167, 8A20 = 170, 8B20 = 171, 8C20 = 172, 8D20 = 173, 8E20 = 174, 8F20 = 175, 8H20 = 177, 8I20 = 178, 8J20 = 179, B220 = 222, B320 = 223, B420 = 224, B720 = 227, B820 = 228, BA20 = 230, BD20 = 233, BE20 = 234, BF20 = 235, BH20 = 237, BI20 = 238, BJ20 = 239, C220 = 242, C420 = 244, C620 = 246, C820 = 248, C920 = 249, CA20 = 250, CB20 = 251, CD20 = 253, CE20 = 254, CF20 = 255, CH20 = 257, CI20 = 258, CJ20 = 259, D220 = 262, D320 = 263, D620 = 266, D720 = 267, DA20 = 270, DB20 = 271, DC20 = 272, DF20 = 275, DH20 = 277, DJ20 = 279, E220 = 282, E320 = 283, E420 = 284, E620 = 286, E720 = 287, E820 = 288, E920 = 289, EC20 = 292, EF20 = 295, EH20 = 297, EI20 = 298, EJ20 = 299, F220 = 302, F320 = 303, F420 = 304, F620 = 306, F820 = 308, F920 = 309, FA20 = 310, FB20 = 311, FD20 = 313, FE20 = 314, FH20 = 317, FI20 = 318, FJ20 = 319, G220 = 322, G920 = 329, GA20 = 330, GB20 = 331, GC20 = 332, GF20 = 335, GH20 = 337, GI20 = 338, GJ20 = 339, H220 = 342, H320 = 343, H420 = 344, H620 = 346, H720 = 347, H820 = 348, H920 = 349, HA20 = 350, HB20 = 351, HC20 = 352, HD20 = 353, HE20 = 354, HI20 = 358, HJ20 = 359, I220 = 362, I320 = 363, I620 = 366, I720 = 367, I920 = 369, IA20 = 370, IB20 = 371, IC20 = 372, ID20 = 373, IE20 = 374, IF20 = 375, J220 = 382, J320 = 383, J420 = 384, J620 = 386, J720 = 387, J820 = 388, J920 = 389, 18620 = 566, 18720 = 567, 18A20 = 570, 18C20 = 572, 19420 = 584, 19820 = 588, 19C20 = 592, 19I20 = 598, 1A320 = 603, 1A420 = 604, 1A820 = 608, 1AC20 = 612, 1AD20 = 613, 1AE20 = 614, 1AF20 = 615, 1AH20 = 617, 1AI20 = 618, 1B320 = 623, 1B420 = 624, 1B620 = 626, 1B720 = 627, 1B820 = 628, 1BA20 = 630, 1BE20 = 634, 1BH20 = 637, 1BI20 = 638, 1C420 = 644, 1CE20 = 654, 1CH20 = 657, 1CI20 = 658, 1D320 = 663, 1D620 = 666, 1D720 = 667, 1D820 = 668, 1DA20 = 670, 1DC20 = 672, 1DI20 = 678, 1E320 = 683, 1E420 = 684, 1E620 = 686, 1E720 = 687, 1E820 = 688, 1EC20 = 692, 1ED20 = 693, 1EF20 = 695, 1F720 = 707, 1F820 = 708, 1FA20 = 710, 1FC20 = 712, 1FD20 = 713, 1FH20 = 717, 1FI20 = 718, 1G220 = 722, 1G820 = 728, 1GA20 = 730, 1GC20 = 732, 1GD20 = 733, 1GF20 = 735, 1H220 = 742, 1H420 = 744, 1H620 = 746, 1H720 = 747, 1H820 = 748, 1HC20 = 752, 1HI20 = 758, 1I220 = 762, 1I720 = 767, 1I820 = 768, 1IA20 = 770, 1IC20 = 772, 1IH20 = 777, 1J220 = 782, 1J420 = 784, 1J620 = 786, 1J720 = 787, 1J820 = 788, 1JA20 = 790, 1JC20 = 792, 1JD20 = 793, 20620 = 806, 20820 = 808, 20920 = 809, 20H20 = 817, 20I20 = 818, 20J20 = 819, 21320 = 823, 21620 = 826, 21720 = 827, 21820 = 828, 21C20 = 832, 21D20 = 833, 21E20 = 834, 21H20 = 837, 21I20 = 838, 23620 = 866, 23720 = 867, 23920 = 869, 23A20 = 870, 23B20 = 871, 23C20 = 872, 23D20 = 873, 23E20 = 874, 23I20 = 878, 23J20 = 879, 24H20 = 897, 24J20 = 899, 26820 = 928, 26920 = 929, 26C20 = 932, 26D20 = 933, 26E20 = 934, 26F20 = 935, 26H20 = 937, 26I20 = 938, 27620 = 946, 27820 = 948, 27920 = 949, 27A20 = 950, 27B20 = 951, 27C20 = 952, 27E20 = 954, 27F20 = 955, 27H20 = 957, 27I20 = 958, 28620 = 966, 28720 = 967, 28A20 = 970, 28C20 = 972, 28D20 = 973, 28E20 = 974, 28F20 = 975, 28H20 = 977, 28I20 = 978, 29420 = 984, 29C20 = 992, 29E20 = 994, 29I20 = 998, 29J20 = 999

 

I cubi esclusionari (o numeri esclusionari del terzo ordine) minori di 106 in base 10 sono: 23 = 8, 33 = 27, 73 = 343, 83 = 512, 273 = 19683, 433 = 79507, 473 = 103823, 483 = 110592, 523 = 140608, 533 = 148877, 633 = 250047, 683 = 314432, 923 = 778688, 1573 = 3869893, 1723 = 5088448, 1873 = 6539203, 1923 = 7077888, 2633 = 18191447, 3783 = 54010152, 4083 = 67917312, 4233 = 75686967, 4583 = 96071912, 4683 = 102503232, 4783 = 109215352, 4873 = 115501303, 5273 = 146363183, 5873 = 202262003, 6083 = 224755712, 6483 = 272097792, 6923 = 331373888, 8233 = 557441767, 8433 = 599077107, 9183 = 773620632, 14573 = 3092990993, 15873 = 3996969003, 15923 = 4034866688, 46573 = 100999381393, 47323 = 105958111168, 56923 = 184414333888, 60583 = 222324747112, 63783 = 259449922152, 76583 = 449103134312.

 

I biquadrati esclusionari in base 10 sono: 2, 3, 4, 7, 8, 9, 24, 27, 28, 32, 42, 52, 53, 58, 59, 67, 89, 93, 203, 258, 284, 324, 329, 832, 843, 2673.

 

In base 10 non esistono quinte potenze esclusionarie.

 

In base 10 il massimo intero esclusionario per i primi ordini è: 639172, 7658, 2673, 0, 92, 93, 712, 0, 18, 12, 4, 0, 37, 0, 9, 0, 0, 3, 4, 0, 7, 2, 7, 0, 8, 3, 9, 0, 0, 0, 0, 0, 2, 2, 2, 0, 0, 2, 0, 0, 2.

 

I numeri piccoli sono spesso esclusionari di diversi ordini; per esempio, 2 è esclusionario di vari ordini in base 10: 22 = 4, 23 = 8, 24 = 16, 26 = 64, 212 = 4096, 214 = 16384, 216 = 65536, 220 1048576, 222 = 4194304, 223 = 8388608, 226 = 67108864, 234 = 17179869184, 235 = 34359738368, 236 = 68719476736, 239 = 549755813888, 242 = 4398046511104, 246 = 70368744177664, 254 = 18014398509481984, 264 = 18446744073709551616, 274 =18889465931478580854784, 283 = 9671406556917033397649408, 2168 = 374144419156711147060143317175368453031918731001856 e nessun altro ordine inferiore a 1000000 (v. Due).

 

Da notare la caratteristica unica di 69: non è esclusionario, ma il suo quadrato, 4761 e il suo cubo, 328509 insieme sono formati dalle 10 cifre, usate tutte una sola volta; è l’unico numero con questa caratteristica in tutte le basi fino a 20.

Bibliografia

  • Pickover, Clifford A.;  A Passion for Mathematics, Hoboken, John Wiley & Sons, 2005.

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