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Esamorfi (numeri)

Rappresentazione dei numeri 

Pickover propose di chiamare esamorfi i numeri naturali che rappresentati in notazione decimale costituiscono la parte finale del corrispondente numero esagonale En, come 26, perché E26 = 1326.

Pickover dimostrò che sono infiniti notando che se k = 5(0)n1, Ek = 5(0)n15(0)n1, dove (d)n rappresenta n ripetizioni della cifra d. Per esempio, E5001 = 50015001.

Più in generale in ogni base pari 2k sono esamorfi tutti i numeri della forma k(0)n e k(0)n1.

Sono invece molto meno comuni i numeri esamorfi in basi dispari: tra le basi inferiori a 20, esistono numeri esamorfi non banali solo in base 15.

 

I numeri esamorfi di k cifre terminano con un numero esamorfo di k – 1 cifre, quindi se non ne esistono di una cifra, non ne esiste nessuno.

 

La tabella riporta i numeri esamorfi inferiori a 109 nelle basi fino a 20, escludendo i casi banali 0 e 1.

Base

Numeri esamorfi

2

102 = 2, 112 = 3, 1002 = 4, 1052 = 5, 10002 = 8, 10012 = 9, 100002 = 16, 100012 = 17, 1000002 = 32, 1000012 = 33, 10000002 = 64, 10000012 = 65, 100000002 = 128, 100000012 = 129, 1000000002 = 256, 1000000012 = 257, 10000000002 = 512, 10000000012 = 513, 100000000002 = 1024, 100000000012 = 1025, 1000000000002 = 2048, 1000000000012 = 2049, 10000000000002 = 4096, 10000000000012 = 4097, 100000000000002 = 8192, 100000000000012 = 8193, 1000000000000002 = 16384, 1000000000000012 = 16385, 10000000000000002 = 32768, 10000000000000012 = 32769, 100000000000000002 = 65536, 100000000000000012 = 65537, 1000000000000000002 = 131072, 1000000000000000012 = 131073, 10000000000000000002 = 262144, 10000000000000000012 = 262145, 100000000000000000002 = 524288, 100000000000000000012 = 524289, 1000000000000000000002 = 1048576, 1000000000000000000012 = 1048577, 10000000000000000000002 = 2097152, 10000000000000000000012 = 2097153, 100000000000000000000002 = 4194304, 100000000000000000000012 = 4194305, 1000000000000000000000002 = 8388608, 1000000000000000000000012 = 8388609, 10000000000000000000000002 = 16777216, 10000000000000000000000012 = 16777217, 100000000000000000000000002 = 33554432, 100000000000000000000000012 = 33554433, 1000000000000000000000000002 = 67108864, 1000000000000000000000000012 = 67108865, 10000000000000000000000000002 = 134217728, 10000000000000000000000000012 = 134217729, 100000000000000000000000000002 = 268435456, 100000000000000000000000000012 = 268435457, 1000000000000000000000000000002 = 536870912, 1000000000000000000000000000012 = 536870913

3

Nessuno

4

24 = 2, 34 = 3, 204 = 8, 214 = 9, 2004 = 32, 2014 = 33, 20004 = 128, 20014 = 129, 200004 = 512, 200014 = 513, 2000004 = 2048, 2000014 = 2049, 20000004 = 8192, 20000014 = 8193, 200000004 = 32768, 200000014 = 32769, 20000014 = 8193, 2000000004 = 131072, 2000000014 = 131073, 20000000004 = 524288, 20000000014 = 524289, 20000000004 = 2097152, 20000000014 = 2097153, 200000000004 = 8388608, 200000000014 = 8388609, 2000000000004 = 33554432, 2000000000014 = 33554433, 20000000000004 = 134217728, 20000000000014 = 134217729, 200000000000004 = 536870912, 200000000000014 = 536870913

5

Nessuno

6

36 = 3, 46 = 4, 136 = 9, 146 = 10, 306 = 18, 316 = 19, 436 = 27, 446 = 28, 2136 = 81, 3006 = 108, 3016 = 109, 3446 = 136, 5136 = 189, 23446 = 568, 30006 = 648, 30016 = 649, 32136 = 729, 53446 = 1216, 202136 = 2673, 300006 = 3888, 300016 = 3889, 353446 = 5104, 502136 = 6561, 2053446 = 16768, 3000006 = 23328, 3000016 = 23329, 3502136 = 29889, 5053446 = 40096, 12053446 = 63424, 13502136 = 76545, 30000006 = 139968, 30000016 = 139969, 42053446 = 203392, 53502136 = 216513, 213502136 = 636417, 300000006 = 839808, 300000016 = 839809, 342053446 = 1043200, 513502136 = 1476225, 2213502136 = 3995649, 3000000006 = 5038848, 3000000016 = 5038849, 3342053446 = 6082048, 5213502136 = 9034497, 22213502136 = 24151041, 30000000006 = 30233088, 30000000016 = 30233089, 33342053446 = 36315136, 52213502136 = 54384129, 222213502136 = 145083393, 300000000006 = 181398528, 300000000016 = 181398529, 333342053446 = 217713664, 522213502136 = 326481921, 1033342053446 = 399112192, 1522213502136 = 689278977

7

Nessuno

8

48 = 4, 58 = 5, 408 = 32, 418 = 33, 4008 = 256, 4018 = 257, 40008 = 2048, 40018 = 2049, 400008 = 16384, 400018 = 16385, 4000008 = 131072, 4000018 = 131072, 40000008 = 1048576, 40000018 = 1048577, 400000008 = 8388608, 400000018 = 8388609, 4000000008 = 67108864, 4000000018 = 67108865, 40000000008 = 536870912, 40000000018 = 536870913

9

Nessuno

10

5, 6, 25, 26, 50, 51, 75, 76, 125, 376, 500, 501, 625, 876, 4376, 5000, 5001, 5625, 9376, 40625, 50000, 50001, 59376, 90625, 109376, 390625, 500000, 500001, 609376, 890625, 2109376, 2890625, 5000000, 5000001, 7109376, 7890625, 12890625, 37109376, 50000000, 50000001, 62890625, 87109376, 212890625, 287109376, 500000000, 500000001, 712890625, 787109376

11

Nessuno

12

312 = 3, 412 = 4, 612 = 6, 712 = 7, 912 = 9, A12 = 10, 5412 = 64, 6012 = 72, 6112 = 73, 6912 = 81, B412 = 136, 25412 = 352, 36912 = 513, 60012 = 864, 60112 = 865, 85412 = 1216, 96912 = 1377, 236912 = 3969, 385412 = 6400, 600012 = 10368, 600112 = 10369, 836912 = 14337, 985412 = 16768, 5385412 = 110080, 6000012 = 124416, 6000112 = 124417, 6836912 = 138753, B385412 = 234496, 1B385412 = 483328, 40836912 = 1009665, 60000012 = 1492992, 60000112 = 1492993, 7B385412 = 1976320, A0836912 = 2502657, 5A0836912 = 17432577, 600000012 = 17915904, 600000112 = 17915905, 61B385412 = 18399232, BA0836912 = 35348481, 561B385412 = 197558272, 6000000012 = 214990848, 6000000112 = 214990849, 65A0836912 = 232423425, B61B385412 = 412549120, 1B61B385412 = 842530816

13

Nessuno

14

714 = 7, 814 = 8, 3714 = 49, 3814 = 50, 7014 = 98, 7114 = 99, A714 = 147, A814 = 148, 1A814 = 344, 53714 = 1029, 70014 = 1372, 70114 = 1373, 8A814 = 1716, C3714 = 2401, 61A814 = 16808, 700014 = 19208, 700114 = 19209, 7C3714 = 21609, D1A814 = 36016, 30C3714 = 117649, 3D1A814 = 151264, 7000014 = 268912, 7000114 = 268913, A0C3714 = 386561, AD1A814 = 420176, 33D1A814 = 1764736, 3A0C3714 = 2000033, 70000014 = 3764768, 70000114 = 3764769, A3D1A814 = 5529504, AA0C3714 = 5764801, 633D1A814 = 46941952, 700000014 = 52706752, 700000114 = 52706753, 7AA0C3714 = 58471553, D33D1A814 = 99648704, 3633D1A814 = 363182464, 37AA0C3714 = 374712065, 7000000014 = 737894528, 7000000114 = 737894529

15

615 = 6, A15 = 10, 6A15 = 100, 8615 = 126, 46A15 = 1000, A8615 = 2376, 146A15 = 4375, DA8615 = 46251, 3146A15 = 156250, BDA8615 = 603126, 4BDA8615 = 3640626, A3146A15 = 7750000, 1A3146A15 = 19140625, D4BDA8615 = 151718751, 4D4BDA8615 = 835156251

16

816 = 8, 916 = 9, 8016 = 128, 8116 = 129, 80016 = 2048, 80116 = 2049, 800016 = 32768, 800116 = 32769, 8000016 = 524288, 8000116 = 524289, 80000016 = 8388608, 80000116 = 8388609, 800000016 = 134217728, 800000116 = 134217729

17

Nessuno

18

918 = 9, A18 = 10, A18 = 10, 4918 = 81, 4A18 = 82, 9018 = 162, 9118 = 163, D918 = 243, DA18 = 244, 24918 = 729, 6DA18 = 2188, 90018 = 2916, 90118 = 2917, B4918 = 3645, FDA18 = 5104, 124918 = 6561, 7FDA18 = 45928, 900018 = 52488, 900118 = 52489, A24918 = 59049, GFDA18 = 98416, 3GFDA18 = 413344, 5124918 = 531441, 9000018 = 944784, 9000118 = 944785, CGFDA18 = 1358128, E124918 = 1476225, 43GFDA18 = 7971616, 4E124918 = 9034497, 90000018 = 17006112, 90000118 = 17006113, D3GFDA18 = 24977728, DE124918 = 26040609, 14E124918 = 43046721, 7D3GFDA18 = 263063296, 900000018 = 306110016, 900000118 = 306110017, A4E124918 = 349156737, GD3GFDA18 = 569173312

19

Nessuno

20

520 = 5, 620 = 6, A20 = 10, B20 = 11, F20 = 15, G20 = 16, 1520 = 25, 8G20 = 176, A020 = 200, A120 = 201, B520 = 225, IG20 = 376, 38G20 = 1376, 6B520 = 2625, A0020 = 4000, A0120 = 4001, D8G20 = 5376, GB520 = 6625, 16B520 = 10625, 8D8G20 = 69376, A00020 = 80000, A00120 = 80001, B6B520 = 90625, ID8G20 = 149376, 98D8G20 = 1509376, A000020 = 1600000, A000120 = 1600001, AB6B520 = 1690625, J8D8G20 = 3109376, 1AB6B520 = 4890625, 898D8G20 = 27109376, A0000020 = 32000000, A0000120 = 32000001, BAB6B520 = 36890625, I98D8G20 = 59109376, 21AB6B520 = 132890625, 7I98D8G20 = 507109376, A00000020 = 640000000, A00000120 = 640000001, C1AB6B520 = 772890625

 

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