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Primi diedri (I)

Rappresentazione dei numeri 

Si chiamano “primi diedri” i numeri primi che restano tali per una permutazione simmetrica delle cifre.

Per capire meglio la definizione, immaginiamo di mettere le cifre ai vertici di un poligono regolare: i primi diedri sono i numeri primi che restano tali per qualsiasi rotazione o riflessione del poligono.

Una definizione alternativa è quella di primi circolari che restano tali anche invertendo l’ordine delle cifre.

 

La tabella seguente mostra i primi diedri di almeno due cifre minori di 109 nelle basi fino a 20.

Base

Primi diedri

2

3 = 112, 7 = 1112, 31 = 111112, 127 = 11111112, 8191 = 11111111111112, 131071 = 111111111111111112, 524287 = 11111111111111111112

3

5 = 123, 7 = 213, 13 = 1113, 1093 = 11111113, 797161 = 11111111111113

4

5 = 114, 134 = 7, 13 = 134, 23 = 1134, 29 = 1314, 53 = 3114, 383 = 113334, 509 = 133314, 863 = 311334, 983 = 331134, 1013 = 333114

5

7 = 125, 11 = 215, 13 = 235, 17 = 325, 19 = 345, 3 = 435, 31 = 1115, 239 = 14245, 359 = 24145, 547 = 41425, 571 = 42415, 1249 = 144445, 2749 = 414445, 3049 = 441445, 3109 = 444145, 3121 = 444415, 19531 = 11111115, 12207031 = 111111111115, 305175781 = 11111111111115

6

7 = 116, 11 = 156, 31 = 516, 43 = 1116, 71 = 1556, 191 = 5156, 211 = 5516, 55987 = 11111116

7

11 = 147, 13 = 167, 17 = 237, 19 = 257, 23 = 327, 29 = 417, 37 = 527, 41 = 567, 43 = 617, 47 = 657, 89 = 1557, 97 = 1667, 257 = 5157, 281 = 5517, 307 = 6167, 337 = 6617, 571 = 14447, 853 = 23267, 997 = 26237, 1021 = 26567, 1171 = 32627, 1453 = 41447, 1579 = 44147, 1597 = 44417, 1657 = 45557, 1951 = 54557, 1993 = 55457, 1999 = 55547, 2029 = 56267, 2179 = 62327, 2203 = 62657, 2347 = 65627, 2801 = 111117, 3001 = 115157, 4177 = 151157, 4201 = 151517, 12433 = 511517, 12601 = 515117, 23557 = 1254527, 36979 = 2125457, 47251 = 2545217, 79987 = 4521257, 89317 = 5212547, 95461 = 5452127

8

13 = 158, 29 = 358, 31 = 378, 41 = 518, 43 = 538, 47 = 578, 59 = 738, 61 = 758, 73 = 1118, 719 = 13178, 971 = 17138, 1657 = 31718, 1759 = 33378, 1787 = 33738, 2011 = 37338, 3673 = 71318, 3803 = 73338

9

11 = 129, 13 = 149, 17 = 189, 19 = 219, 23 = 259, 37 = 419, 43 = 479, 47 = 529, 67 = 749, 71 = 789, 73 = 819, 79 = 879, 101 = 1229, 173 = 2129, 181 = 2219, 647 = 7889, 719 = 8789, 727 = 8879, 3371 = 45559, 4019 = 54559, 4091 = 55459, 4099 = 55549, 5741 = 77789, 5749 = 77879, 5821 = 78779, 6469 = 87779, 37087 = 557779, 38543 = 577759, 50047 = 755779, 51487 = 775579, 51647 = 777559, 3109823 = 57587789, 3181327 = 58778579, 4073567 = 75877859, 4190183 = 77857589, 4230871 = 78575879, 4597063 = 85758779, 4717103 = 87785759

10

11, 13, 17, 31, 37, 71, 73, 79, 97, 113, 131, 199, 311, 337, 373, 733, 919, 991, 11939, 19391, 39119, 91193, 93911, 193939, 391939, 393919, 919393, 939193, 939391

11

1211 = 13, 1611 = 17, 1811 = 19, 23 = 2111, 29 = 2711, 31 = 2911, 37 = 3411, 43 = 3A11, 47 = 4311, 53 = 4911, 61 = 5611, 67 = 6111, 71 = 6511, 73 = 6711, 79 = 7211, 83 = 7611, 89 = 8111, 97 = 8911, 101 = 9211, 103 = 9411, 107 = 9811, 113 = A311, 139 = 11711, 163 = 13911, 199 = 17111, 223 = 19311, 241 = 1AA11, 383 = 31911, 401 = 33511, 421 = 35311, 439 = 36A11, 463 = 39111, 479 = 3A611, 641 = 53311, 677 = 56611, 701 = 58811, 769 = 63A11, 787 = 65611, 797 = 66511, 839 = 6A311, 859 = 71111, 967 = 7AA11, 1031 = 85811, 1061 = 88511, 1103 = 91311, 1123 = 93111, 1231 = A1A11, 1249 = A3611, 1279 = A6311, 1297 = A7A11, 1321 = AA111, 1327 = AA711, 3373 = 259711, 3593 = 277711, 3613 = 279511, 4591 = 34A411, 4639 = 353811, 4957 = 37A711, 4999 = 383511, 5279 = 3A6A11, 5741 = 434A11, 6581 = 4A4311, 6983 = 527911, 7109 = 538311, 7823 = 597211, 9239 = 6A3A11, 9623 = 725911, 9643 = 727711, 9767 = 737A11, 10193 = 772711, 10243 = 777211, 10463 = 795211, 10607 = 7A7311, 11069 = 835311, 12613 = 952711, 12853 = 972511, 13309 = 9AAA11, 13789 = A3A611, 13831 = A43411, 14149 = A6A311, 14197 = A73711, 14519 = A9AA11, 14629 = AA9A11, 14639 = AAA911, 27697 = 1989A11, 29137 = 1A98911, 117127 = 7AAAA11, 129347 = 891A911, 130337 = 89A1911, 134417 = 91A9811, 143527 = 9891A11, 143617 = 989A111, 145307 = 9A19811, 148927 = A198911, 157057 = A7AAA11, 159457 = A989111, 160687 = AA7AA11, 161017 = AAA7A11, 161047 = AAAA711, 221567 = 14151511, 229739 = 14767411, 236087 = 15141511, 236207 = 15151411, 665089 = 41476711, 665677 = 41515111, 755569 = 47674111, 767831 = 48497911, 786431 = 49794811, 825397 = 51415111, 826717 = 51514111, 1074289 = 67414711, 1187819 = 74147611, 1225019 = 76741411, 1265471 = 79484911, 1359901 = 84979411, 1519261 = 94849711, 1564501 = 97948411, 49544447 = 25A6A52711, 51784727 = 2725A6A511, 102143557 = 52725A6A11, 116271157 = 5A6A527211, 135482797 = 6A52725A11, 140914237 = 725A6A5211, 204157487 = A52725A611, 207188327 = A6A5272511

12

13 = 1112, 17 = 1512, 61 = 5112, 67 = 5712, 71 = 5B12, 89 = 7512, 137 = B512, 157 = 11112, 163 = 11712, 167 = 11B12, 229 = 17112, 277 = 1B112, 1021 = 71112, 1597 = B1112, 9431 = 555B12, 9497 = 55B512, 10289 = 5B5512, 19793 = B55512, 22621 = 1111112

13

17 = 1413, 19 = 1613, 23 = 1A13, 29 = 2313, 31 = 2513, 41 = 3213, 47 = 3813, 53 = 4113, 67 = 5213, 71 = 5613, 73 = 5813, 79 = 6113, 83 = 6513, 89 = 6B13, 101 = 7A13, 103 = 7C13, 107 = 8313, 109 = 8513, 127 = 9A13, 131 = A113, 137 = A713, 139 = A913, 149 = B613, 163 = C713, 193 = 11B13, 211 = 13313, 239 = 15513, 313 = 1B113, 373 = 22913, 397 = 24713, 433 = 27413, 457 = 29213, 523 = 31313, 547 = 33113, 557 = 33B13, 619 = 38813, 653 = 3B313, 709 = 42713, 769 = 47213, 863 = 51513, 911 = 55113, 1213 = 72413, 1237 = 74213, 1283 = 77913, 1297 = 78A13, 1307 = 79713, 1321 = 7A813, 1399 = 83813, 1453 = 87A13, 1459 = 88313, 1489 = 8A713, 1549 = 92213, 1619 = 97713, 1789 = A7813, 1801 = A8713, 1873 = B1113, 1901 = B3313, 2729 = 131C13, 3373 = 16C613, 3571 = 181913, 3739 = 191813, 4241 = 1C1313, 5113 = 243413, 5783 = 282B13, 6211 = 2A9A13, 6287 = 2B2813, 6917 = 31C113, 7297 = 342413, 8111 = 38CC13, 8783 = 3CC813, 9181 = 424313, 9349 = 434213, 9521 = 444513, 9533 = 445413, 9689 = 454413, 11717 = 544413, 12241 = 575813, 12409 = 585713, 13441 = 616C13, 14281 = 666713, 14293 = 667613, 14449 = 676613, 15131 = 6B6C13, 15289 = 6C6113, 15299 = 6C6B13, 16333 = 758513, 16477 = 766613, 17863 = 819113, 18059 = 82B213, 18251 = 83CC13, 18517 = 857513, 19763 = 8CC313, 20047 = 918113, 21499 = 9A2A13, 22447 = A2A913, 23623 = A9A213, 24611 = B28213, 25343 = B6C613, 26573 = C13113, 26987 = C38C13, 27397 = C61613, 27527 = C6B613, 27767 = C83C13, 28439 = CC3813, 28499 = CC8313, 30941 = 1111113, 97561 = 3553913, 106033 = 3935513, 150961 = 5393513, 154417 = 5539313, 264553 = 9355313, 1812263 = 4B5B5B13, 2181359 = 5B4B5B13, 2183543 = 5B5B4B13, 4223627 = B4B5B513, 4252019 = B5B4B513, 4252187 = B5B5B413, 5229043 = 111111113

14

17 = 1314, 19 = 1514, 23 = 1914, 43 = 3114, 47 = 3514, 53 = 3B14, 71 = 5114, 73 = 5314, 79 = 5914, 127 = 9114, 131 = 9514, 137 = 9B14, 139 = 9D14, 157 = B314, 163 = B914, 167 = BD14, 191 = D914, 193 = DB14, 211 = 11114, 643 = 33D14, 773 = 3D314, 2593 = D3314, 2963 = 111914, 3067 = 119114, 3911 = 15D514, 4523 = 191114, 13999 = 515D14, 16339 = 5D5114, 24907 = 911114, 36671 = D51514, 733651 = 15151914, 734431 = 15191514, 887311 = 19151514, 2741579 = 51519114, 2752499 = 51915114, 4892819 = 91515114, 8108731 = 111111114

15

17 = 1215, 1415 = 19, 29 = 1E15, 31 = 2115, 37 = 2715, 41 = 2B15, 43 = 2D15, 61 = 4115, 67 = 4715, 73 = 4D15, 107 = 7215, 109 = 7415, 113 = 7815, 127 = 8715, 131 = 8B15, 167 = B215, 173 = B815, 197 = D215, 199 = D415, 211 = E115, 241 = 11115, 433 = 1DD15, 487 = 22715, 557 = 27215, 971 = 44B15, 1069 = 4B415, 1607 = 72215, 1693 = 77D15, 1777 = 7D715, 2539 = B4415, 2699 = BEE15, 2953 = D1D15, 3037 = D7715, 3121 = DD115, 3329 = EBE15, 3371 = EEB15, 4297 = 141715, 4339 = 144415, 4969 = 171415, 9341 = 2B7B15, 13789 = 414415, 13831 = 417115, 14419 = 441415, 14461 = 444115, 15473 = 48B815, 23911 = 714115, 26141 = 7B2B15, 28031 = 848B15, 29599 = 8B8415, 37747 = B2B715, 38867 = B7B215, 38993 = B84815, 379693 = 7777D15, 379777 = 777D715, 380591 = 77B7B15, 381037 = 77D7715, 383291 = 7887B15, 393191 = 7B77B15, 393203 = 7B78815, 393247 = 7B7B715, 399937 = 7D77715, 431213 = 87B7815, 433747 = 887B715, 582247 = B77B715, 582427 = B788715, 583087 = B7B7715, 683437 = D777715

16

17 = 1116, 23 = 1716, 31 = 1F16, 53 = 3516, 59 = 3B16, 61 = 3D16, 83 = 5316, 89 = 5916, 113 = 7116, 149 = 9516, 179 = B316, 191 = BF16, 211 = D316, 241 = F116, 251 = FB16, 277 = 11516, 283 = 11B16, 337 = 15116, 433 = 1B116, 887 = 37716, 1297 = 51116, 1373 = 55D16, 1493 = 5D516, 1847 = 73716, 1907 = 77316, 1979 = 7BB16, 2833 = B1116, 2939 = B7B16, 2999 = BB716, 3037 = BDD16, 3413 = D5516, 3517 = DBD16, 3547 = DDB16, 7549 = 1D7D16, 14143 = 373F16, 14713 = 397916, 16183 = 3F3716, 29683 = 73F316, 31033 = 793916, 32029 = 7D1D16, 37783 = 939716, 38803 = 979316, 53719 = D1D716, 55249 = D7D116, 62323 = F37316, 129439 = 1F99F16, 630559 = 99F1F16, 651769 = 9F1F916, 991129 = F1F9916, 1022449 = F99F116, 1940893 = 1D9D9D16, 10296733 = 9D1D9D16, 10329373 = 9D9D1D16, 13752793 = D1D9D916, 14275033 = D9D1D916, 14277073 = D9D9D116

17

23 = 1617, 31 = 1E17, 37 = 2317, 47 = 2D17, 53 = 3217, 59 = 3817, 61 = 3A17, 73 = 4517, 79 = 4B17, 89 = 5417, 101 = 5G17, 103 = 6117, 113 = 6B17, 131 = 7C17, 139 = 8317, 149 = 8D17, 151 = 8F17, 163 = 9A17, 173 = A317, 179 = A917, 181 = AB17, 191 = B417, 193 = B617, 197 = BA17, 211 = C717, 223 = D217, 229 = D817, 239 = E117, 263 = F817, 277 = G517, 307 = 11117, 2203 = 7AA17, 2311 = 7GG17, 3019 = A7A17, 3067 = AA717, 4759 = G7G17, 4903 = GG717, 5519 = 121B17, 6277 = 14C417, 6311 = 14E417, 7949 = 1A8A17, 8111 = 1B1217, 10303 = 21B117, 17981 = 3B3C17, 18269 = 3C3B17, 20021 = 414C17, 20023 = 414E17, 23189 = 4C4117, 23609 = 4DBD17, 23767 = 4E4117, 31907 = 686F17, 33469 = 6DDD17, 33923 = 6F6817, 38723 = 7EGE17, 41299 = 86F617, 42221 = 8A1A17, 49597 = A1A817, 51613 = A8A117, 54367 = B12117, 55117 = B3C317, 57881 = BD4D17, 60013 = C3B317, 60133 = C41417, 65257 = D4DB17, 65837 = D6DD17, 67273 = DBD417, 67741 = DD6D17, 67853 = DDD617, 69959 = E41417, 71059 = E7EG17, 73651 = EGE717, 75571 = F68617, 82787 = GE7E17, 88741 = 1111117, 1538087 = 11711C17, 1562647 = 11C11717, 2009911 = 1711C117, 2012641 = 171B2B17, 2076421 = 17EAE717, 2351617 = 1B2B1717, 2427431 = 1C117117, 3765409 = 2B171B17, 3856261 = 2C2F7F17, 4131301 = 2F7F2C17, 8046811 = 5B5EAE17, 8321851 = 5EAE5B17, 10030919 = 711C1117, 10061141 = 717EAE17, 10077329 = 71B2B117, 11161589 = 7EAE7117, 11205157 = 7F2C2F17, 15395707 = AE5B5E17, 15402677 = AE717E17, 15736817 = B171B217, 15839921 = B2B17117, 16107947 = B5EAE517, 17128759 = C1171117, 17281301 = C2F7F217, 20351339 = E5B5EA17, 20469829 = E717EA17, 20783627 = EAE5B517, 20784037 = EAE71717, 21524693 = F2C2F717, 21956981 = F7F2C217, 25646167 = 111111117

18

19 = 1118, 29 = 1B18, 97 = 5718, 103 = 5D18, 107 = 5H18, 131 = 7518, 139 = 7D18, 199 = B118, 239 = D518, 241 = D718, 311 = H518, 353 = 11B18, 359 = 11H18, 523 = 1B118, 571 = 1DD18, 631 = 1H118, 1721 = 55B18, 1823 = 5B518, 2477 = 7BB18, 2591 = 7HH18, 3583 = B1118, 3659 = B5518, 3701 = B7B18, 3769 = BB718, 3779 = BBH18, 3881 = BHB18, 4243 = D1D18, 4447 = DD118, 5527 = H1118, 5651 = H7H18, 5717 = HBB18, 5821 = HH718, 7481 = 151B18, 9419 = 1B1518, 9431 = 1B1H18, 11369 = 1H1B18, 29683 = 51B118, 32969 = 5BDB18, 64567 = B15118, 64783 = B1H118, 65983 = B5BD18, 68567 = BDB518, 68611 = BDDD18, 68683 = BDHD18, 79481 = DB5B18, 79627 = DBDD18, 79631 = DBDH18, 80239 = DDBD18, 80273 = DDDB18, 81569 = DHDB18, 99667 = H1B118, 103567 = HDBD18

19

29 = 1A19, 31 = 1C19, 41 = 2319, 43 = 2519, 47 = 2919, 59 = 3219, 61 = 3419, 67 = 3A19, 71 = 3E19, 73 = 3G19, 79 = 4319, 83 = 4719, 89 = 4D19, 97 = 5219, 103 = 5819, 107 = 5C19, 109 = 5E19, 113 = 5I19, 137 = 7419, 149 = 7G19, 151 = 7I19, 157 = 8519, 167 = 8F19, 173 = 9219, 181 = 9A19, 191 = A119, 193 = A319, 199 = A919, 223 = BE19, 227 = BI19, 229 = C119, 233 = C519, 251 = D419, 263 = DG19, 269 = E319, 271 = E519, 277 = EB19, 283 = EH19, 293 = F819, 307 = G319, 311 = G719, 317 = GD19, 337 = HE19, 347 = I519, 349 = I719, 353 = IB19, 383 = 11319, 401 = 12219, 419 = 13119, 601 = 1CC19, 661 = 1FF19, 743 = 21219, 761 = 22119, 773 = 22D19, 907 = 29E19, 971 = 2D219, 997 = 2E919, 1103 = 31119, 1223 = 37719, 1283 = 3AA19, 1531 = 44B19, 1657 = 4B419, 2591 = 73719, 2663 = 77319, 2677 = 77H19, 2857 = 7H719, 3301 = 92E19, 3517 = 9E219, 3677 = A3A19, 3803 = AA319, 4051 = B4419, 4271 = BFF19, 4363 = C1C19, 4561 = CC119, 4733 = D2219, 5101 = E2919, 5227 = E9219, 5449 = F1F19, 5639 = FBF19, 5701 = FF119, 5711 = FFB19, 6277 = H7719, 9697 = 17G719, 13679 = 1IGI19, 15569 = 252819, 16649 = 282519, 17033 = 293919, 18671 = 2DDD19, 22093 = 343F19, 23873 = 392919, 24979 = 3C3D19, 25339 = 3D3C19, 26053 = 3F3419, 28807 = 43F319, 31151 = 4A5A19, 35171 = 528219, 37781 = 59C919, 37991 = 5A4A19, 39953 = 5FCF19, 48523 = 717G19, 51767 = 7A7B19, 52127 = 7B7A19, 53923 = 7G7119, 55691 = 825219, 60149 = 8EBE19, 62627 = 929319, 62987 = 939219, 63719 = 959C19, 66239 = 9C9519, 67427 = 9FEF19, 70229 = A4A519, 70589 = A5A419, 71333 = A7B719, 78173 = B7A719, 80669 = BE8E19, 83641 = C3D319, 85661 = C95919, 87833 = CF5F19, 90149 = D2DD19, 90481 = D3C319, 93911 = DD2D19, 94109 = DDD219, 99191 = E8EB19, 100271 = EBE819, 101627 = EF9F19, 104047 = F34319, 104987 = F5FC19, 106433 = F9FE19, 107507 = FCF519, 108233 = FEF919, 112297 = G71719, 116279 = GI1I19, 124181 = I1IG19, 129581 = IGI119, 234361 = 1F33F19, 416417 = 33D9D19, 416989 = 33F1F19, 483629 = 3D9D319, 494497 = 3F1F319, 1238129 = 999DD19, 1239569 = 99DD919, 1263209 = 9D33D19, 1266929 = 9DD9919, 1716089 = D33D919, 1759337 = D999D19, 1760657 = D9D3319, 1786769 = DD99919, 1967149 = F1F3319, 1976761 = F33F119, 3492967 = 17F4F719, 5241043 = 22422719, 5261617 = 22722419, 5488057 = 24227219, 5878963 = 27224219, 6609601 = 2CDC2E19, 6795001 = 2E2CDC19, 9279007 = 3E3FBF19, 9464047 = 3FBF3E19, 10181323 = 42272219, 11907733 = 4F717F19, 13391401 = 57E75B19, 13594309 = 595I6I19, 13851121 = 5B57E719, 14774029 = 5I6I5919, 17240029 = 6I595I19, 17516533 = 717F4F19, 17608537 = 72242219, 18061699 = 75B57E19, 19207219 = 7E75B519, 19320493 = 7F4F7119, 23062471 = 95I6I519, 27941899 = B57E7519, 29217607 = BF3E3F19, 30070819 = C2E2CD19, 31490659 = CDC2E219, 33771961 = DC2E2C19, 35013259 = E2CDC219, 35163493 = E3FBF319, 35616001 = E75B5719, 37629853 = F3E3FB19, 37768207 = F4F71719, 38063407 = F717F419, 38679253 = FBF3E319, 45285271 = I595I619, 45477151 = I6I59519, 150908467 = 33HI9IH19, 185645659 = 3HI9IH319, 470220169 = 9IH33HI19, 807722527 = H33HI9I19, 845652307 = HI9IH3319, 871574287 = I9IH33H19, 889337569 = IH33HI919

20

23 = 1320, 29 = 1920, 61 = 3120, 71 = 3B20, 73 = 3D20, 79 = 3J20, 151 = 7B20, 157 = 7H20, 181 = 9120, 191 = 9B20, 193 = 9D20, 197 = 9H20, 199 = 9J20, 223 = B320, 227 = B720, 229 = B920, 233 = BD20, 263 = D320, 269 = D920, 271 = DB20, 277 = DH20, 347 = H720, 349 = H920, 353 = HD20, 359 = HJ20, 383 = J320, 389 = J920, 397 = JH20, 421 = 11120, 1277 = 33H20, 1543 = 3H320, 2953 = 77D20, 3067 = 7D720, 5347 = D7720, 6863 = H3320, 8423 = 111320, 8429 = 111920, 8461 = 113120, 8581 = 119120, 9221 = 131120, 11621 = 191120, 12437 = 1B1H20, 14831 = 1H1B20, 24421 = 311120, 72421 = 911120, 88741 = B1H120, 136621 = H1B120, 5112791 = 1BJ1JB20, 6328639 = 1JB1BJ20, 35455639 = B1BJ1J20, 38255821 = BJ1JB120, 61116431 = J1JB1B20, 62572781 = JB1BJ120

 

Gli unici primi diedri in base 2 sono i primi pluriunitari, cioè i primi di Mersenne.

 

In qualsiasi base tutti i primi permutabili sono primi diedri; oltre a questi se ne conoscono pochissimi: in base 10 gli unici noti sono 11939 e 193939, oltre a quelli ottenuti con le varie permutazioni di cifre.

 

La tabella seguente mostra i primi diedri non permutabili di almeno due cifre minori di 109 nelle basi fino a 20.

Base

Primi diedri non permutabili

2

-

3

-

4

83 = 113334, 509 = 133314, 863 = 311334, 983 = 331134, 1013 = 333114

5

239 = 14245, 359 = 24145, 547 = 41425, 571 = 42415

6

-

7

853 = 23267, 997 = 26237, 1021 = 26567, 1171 = 32627, 2029 = 56267, 2179 = 62327, 2203 = 62657, 2347 = 65627, 3001 = 115157, 4177 = 151157, 4201 = 151517, 12433 = 511517, 12601 = 515117, 23557 = 1254527, 36979 = 2125457, 47251 = 2545217, 79987 = 4521257, 89317 = 5212547, 95461 = 5452127

8

719 = 13178, 971 = 17138, 1657 = 31718, 3673 = 71318

9

37087 = 557779, 38543 = 577759, 50047 = 755779, 51487 = 775579, 51647 = 777559, 3109823 = 57587789, 3181327 = 58778579, 4073567 = 75877859, 4190183 = 77857589, 4230871 = 78575879, 4597063 = 85758779, 4717103 = 87785759

10

11939, 19391, 39119, 91193, 93911, 193939, 391939, 393919, 919393, 939193, 939391

11

3373 = 259711, 3613 = 279511, 4591 = 34A411, 4639 = 353811, 4957 = 37A711, 4999 = 383511, 5279 = 3A6A11, 5741 = 434A11, 6581 = 4A4311, 6983 = 527911, 7109 = 538311, 7823 = 597211, 9239 = 6A3A11, 9623 = 725911, 9767 = 737A11, 10463 = 795211, 10607 = 7A7311, 11069 = 835311, 12613 = 952711, 12853 = 972511, 13789 = A3A611, 13831 = A43411, 14149 = A6A311, 14197 = A73711, 27697 = 1989A11, 29137 = 1A98911, 129347 = 891A911, 130337 = 89A1911, 134417 = 91A9811, 143527 = 9891A11, 143617 = 989A111, 145307 = 9A19811, 148927 = A198911, 159457 = A989111, 221567 = 14151511, 229739 = 14767411, 236087 = 15141511, 236207 = 15151411, 665089 = 41476711, 665677 = 41515111, 755569 = 47674111, 767831 = 48497911, 786431 = 49794811, 825397 = 51415111, 826717 = 51514111, 1074289 = 67414711, 1187819 = 74147611, 1225019 = 76741411, 1265471 = 79484911, 1359901 = 84979411, 1519261 = 94849711, 1564501 = 97948411, 49544447 = 25A6A52711, 51784727 = 2725A6A511, 102143557 = 52725A6A11, 116271157 = 5A6A527211, 135482797 = 6A52725A11, 140914237 = 725A6A5211, 204157487 = A52725A611, 207188327 = A6A5272511

12

-

13

2729 = 131C13, 3373 = 16C613, 3571 = 181913, 3739 = 191813, 4241 = 1C1313, 5113 = 243413, 5783 = 282B13, 6211 = 2A9A13, 6287 = 2B2813, 6917 = 31C113, 7297 = 342413, 8111 = 38CC13, 8783 = 3CC813, 9181 = 424313, 9349 = 434213, 12241 = 575813, 12409 = 585713, 13441 = 616C13, 15131 = 6B6C13, 15289 = 6C6113, 15299 = 6C6B13, 16333 = 758513, 17863 = 819113, 18059 = 82B213, 18251 = 83CC13, 18517 = 857513, 19763 = 8CC313, 20047 = 918113, 21499 = 9A2A13, 22447 = A2A913, 23623 = A9A213, 24611 = B28213, 25343 = B6C613, 26573 = C13113, 26987 = C38C13, 27397 = C61613, 27527 = C6B613, 27767 = C83C13, 28439 = CC3813, 28499 = CC8313, 97561 = 3553913, 106033 = 3935513, 150961 = 5393513, 154417 = 5539313, 264553 = 9355313, 1812263 = 4B5B5B13, 2181359 = 5B4B5B13, 2183543 = 5B5B4B13, 4223627 = B4B5B513, 4252019 = B5B4B513, 4252187 = B5B5B413

14

3911 = 15D514, 13999 = 515D14, 16339 = 5D5114, 36671 = D51514, 733651 = 15151914, 734431 = 15191514, 887311 = 19151514, 2741579 = 51519114, 2752499 = 51915114, 4892819 = 91515114

15

4297 = 141715, 4969 = 171415, 9341 = 2B7B15, 13831 = 417115, 15473 = 48B815, 23911 = 714115, 26141 = 7B2B15, 28031 = 848B15, 29599 = 8B8415, 37747 = B2B715, 38867 = B7B215, 38993 = B84815, 380591 = 77B7B15, 383291 = 7887B15, 393191 = 7B77B15, 393203 = 7B78815, 393247 = 7B7B715, 431213 = 87B7815, 433747 = 887B715, 582247 = B77B715, 582427 = B788715, 583087 = B7B7715

16

7549 = 1D7D16, 14143 = 373F16, 14713 = 397916, 16183 = 3F3716, 29683 = 73F316, 31033 = 793916, 32029 = 7D1D16, 37783 = 939716, 38803 = 979316, 53719 = D1D716, 55249 = D7D116, 62323 = F37316, 129439 = 1F99F16, 630559 = 99F1F16, 651769 = 9F1F916, 991129 = F1F9916, 1022449 = F99F116, 1940893 = 1D9D9D16, 10296733 = 9D1D9D16, 10329373 = 9D9D1D16, 13752793 = D1D9D916, 14275033 = D9D1D916, 14277073 = D9D9D116

17

5519 = 121B17, 6277 = 14C417, 6311 = 14E417, 7949 = 1A8A17, 8111 = 1B1217, 10303 = 21B117, 17981 = 3B3C17, 18269 = 3C3B17, 20021 = 414C17, 20023 = 414E17, 23189 = 4C4117, 23609 = 4DBD17, 23767 = 4E4117, 31907 = 686F17, 33923 = 6F6817, 38723 = 7EGE17, 41299 = 86F617, 42221 = 8A1A17, 49597 = A1A817, 51613 = A8A117, 54367 = B12117, 55117 = B3C317, 57881 = BD4D17, 60013 = C3B317, 60133 = C41417, 65257 = D4DB17, 67273 = DBD417, 69959 = E41417, 71059 = E7EG17, 73651 = EGE717, 75571 = F68617, 82787 = GE7E17, 1538087 = 11711C17, 1562647 = 11C11717, 2009911 = 1711C117, 2012641 = 171B2B17, 2076421 = 17EAE717, 2351617 = 1B2B1717, 2427431 = 1C117117, 3765409 = 2B171B17, 3856261 = 2C2F7F17, 4131301 = 2F7F2C17, 8046811 = 5B5EAE17, 8321851 = 5EAE5B17, 10030919 = 711C1117, 10061141 = 717EAE17, 10077329 = 71B2B117, 11161589 = 7EAE7117, 11205157 = 7F2C2F17, 15395707 = AE5B5E17, 15402677 = AE717E17, 15736817 = B171B217, 15839921 = B2B17117, 16107947 = B5EAE517, 17128759 = C1171117, 17281301 = C2F7F217, 20351339 = E5B5EA17, 20469829 = E717EA17, 20783627 = EAE5B517, 20784037 = EAE71717, 21524693 = F2C2F717, 21956981 = F7F2C217

18

7481 = 151B18, 9419 = 1B1518, 9431 = 1B1H18, 11369 = 1H1B18, 29683 = 51B118, 32969 = 5BDB18, 64567 = B15118, 64783 = B1H118, 65983 = B5BD18, 68567 = BDB518, 68683 = BDHD18, 79481 = DB5B18, 79631 = DBDH18, 81569 = DHDB18, 99667 = H1B118, 103567 = HDBD18

19

9697 = 17G719, 13679 = 1IGI19, 15569 = 252819, 16649 = 282519, 17033 = 293919, 22093 = 343F19, 23873 = 392919, 24979 = 3C3D19, 25339 = 3D3C19, 26053 = 3F3419, 28807 = 43F319, 31151 = 4A5A19, 35171 = 528219, 37781 = 59C919, 37991 = 5A4A19, 39953 = 5FCF19, 48523 = 717G19, 51767 = 7A7B19, 52127 = 7B7A19, 53923 = 7G7119, 55691 = 825219, 60149 = 8EBE19, 62627 = 929319, 62987 = 939219, 63719 = 959C19, 66239 = 9C9519, 67427 = 9FEF19, 70229 = A4A519, 70589 = A5A419, 71333 = A7B719, 78173 = B7A719, 80669 = BE8E19, 83641 = C3D319, 85661 = C95919, 87833 = CF5F19, 90481 = D3C319, 99191 = E8EB19, 100271 = EBE819, 101627 = EF9F19, 104047 = F34319, 104987 = F5FC19, 106433 = F9FE19, 107507 = FCF519, 108233 = FEF919, 112297 = G71719, 116279 = GI1I19, 124181 = I1IG19, 129581 = IGI119, 234361 = 1F33F19, 416417 = 33D9D19, 416989 = 33F1F19, 483629 = 3D9D319, 494497 = 3F1F319, 1238129 = 999DD19, 1239569 = 99DD919, 1263209 = 9D33D19, 1266929 = 9DD9919, 1716089 = D33D919, 1759337 = D999D19, 1760657 = D9D3319, 1786769 = DD99919, 1967149 = F1F3319, 1976761 = F33F119, 3492967 = 17F4F719, 5241043 = 22422719, 5261617 = 22722419, 5488057 = 24227219, 5878963 = 27224219, 6609601 = 2CDC2E19, 6795001 = 2E2CDC19, 9279007 = 3E3FBF19, 9464047 = 3FBF3E19, 10181323 = 42272219, 11907733 = 4F717F19, 13391401 = 57E75B19, 13594309 = 595I6I19, 13851121 = 5B57E719, 14774029 = 5I6I5919, 17240029 = 6I595I19, 17516533 = 717F4F19, 17608537 = 72242219, 18061699 = 75B57E19, 19207219 = 7E75B519, 19320493 = 7F4F7119, 23062471 = 95I6I519, 27941899 = B57E7519, 29217607 = BF3E3F19, 30070819 = C2E2CD19, 31490659 = CDC2E219, 33771961 = DC2E2C19, 35013259 = E2CDC219, 35163493 = E3FBF319, 35616001 = E75B5719, 37629853 = F3E3FB19, 37768207 = F4F71719, 38063407 = F717F419, 38679253 = FBF3E319, 45285271 = I595I619, 45477151 = I6I59519, 150908467 = 33HI9IH19, 185645659 = 3HI9IH319, 470220169 = 9IH33HI19, 807722527 = H33HI9I19, 845652307 = HI9IH3319, 871574287 = I9IH33H19, 889337569 = IH33HI919

20

12437 = 1B1H20, 14831 = 1H1B20, 88741 = B1H120, 136621 = H1B120, 5112791 = 1BJ1JB20, 6328639 = 1JB1BJ20, 35455639 = B1BJ1J20, 38255821 = BJ1JB120, 61116431 = J1JB1B20, 62572781 = JB1BJ120

 

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