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Illuminati (numeri)

Rappresentazione dei numeri 

Nella filosofia buddista un illuminato è una persona che capisce l’essenza delle cose, incluso se stesso.

Nel 2002 Joseph L. Pe propose di chiamare “illuminati” i numeri composti che comprendono se stessi, perché iniziano con i loro fattori primi, concatenati senza ripetizioni in ordine crescente. Per esempio, sono illuminati 250 = 2 • 53, perché inizia con 2 e 5, e 119911 = 112 • 991, perché inizia con 11, seguito da 991.

 

La definizione si può naturalmente estendere alla rappresentazione in qualsiasi base.

 

In qualsiasi base tutti i numeri illuminati sono multipli di un quadrato.

I numeri illuminati con due soli fattori primi sono i quadrati dei numeri primi, nella base uguale al numero primo, perché p2 in base p si rappresenta come 100p e inizia con 10p, la rappresentazione di p in base p.

 

Tutte le potenze di un primo p sono numeri illuminati in base p.

 

In qualsiasi base b esistono infiniti numeri illuminati della forma pnq, con p maggiore di 1 e non uguale a una potenza con esponente razionale di b e q maggiore di zero (M. Fiorentini, 2018).

 

In base 10:

  • il minimo numero illuminato è 250 = 2 • 53;

  • il minimo numero illuminato dispari è 119911 = 112 • 991;

  • i minimi numeri illuminati quadrati sono 162 = 256 = 28 e 482 = 2304 = 28 • 32;

  • il minimo numero illuminato cubo è 143 = 2744 = 23 • 73.

 

I numeri illuminati in base 10 fino a 1012 sono (Giovanni Resta The Online Encyclopedia of Integer Sequences http://oeis.org):

250 = 2 • 53,

256 = 28,

2048 = 211,

2176 = 27 • 17,

2304 = 28 • 32,

2500 = 22 • 54,

2560 = 29 • 5,

2744 = 23 • 73,

23328 = 25 • 36,

25000 = 23 • 55,

25600 = 210 • 52,

119911 = 112 • 991,

219488 = 25 • 193,

236196 = 22 • 310,

250000 = 24 • 56,

256000 = 211 • 53,

262144 = 218,

290912 = 25 • 9091,

2097152 = 221,

2238728 = 23 • 234,

2317312 = 210 • 31 • 73,

2359296 = 218 • 32,

2370816 = 28 • 33 • 73,

2500000 = 25 • 57,

2560000 = 212 • 54,

3515625 = 32 • 58,

3720087 = 312 • 7,

5117695 = 5 • 113 • 769,

13436683 = 132 • 433,

21359416 = 23 • 13 • 593,

23592960 = 219 • 32 • 5,

23887872 = 215 • 36,

25000000 = 26 • 58,

25600000 = 213 • 55,

27294568 = 23 • 76 • 29,

29090912 = 25 • 909091,

235929600 = 220 • 32 • 52,

237180384 = 25 • 32 • 77,

248543744 = 29 • 485437,

250000000 = 27 • 59,

256000000 = 214 • 56,

268435456 = 228,

271351808 = 215 • 72 • 132,

275365888 = 214 • 75,

353109375 = 36 • 56 • 31,

358722675 = 315 • 52,

387420489 = 318,

595238125 = 54 • 952381,

761743661 = 75 • 61 • 743,

2147483648 = 231,

2317206624 = 25 • 3 • 176,

2324522934 = 2 • 319,

2347906338 = 2 • 312 • 472,

2359296000 = 221 • 32 • 53,

2377970784 = 25 • 35 • 72 • 792,

2378170368 = 222 • 34 • 7,

2500000000 = 28 • 510,

2560000000 = 215 • 57,

2717384704 = 219 • 71 • 73,

3486784401 = 320,

3573588375 = 35 • 53 • 76,

21109276672 = 217 • 115,

21923380546 = 2 • 196 • 233,

22707961856 = 223 • 2707,

23219011584 = 217 • 311,

23573790720 = 215 • 33 • 5 • 732,

23592960000 = 222 • 32 • 54,

23679672516 = 22 • 39 • 673,

23702740992 = 213 • 310 • 72,

23727920916 = 22 • 3 • 711,

24312841216 = 210 • 432 • 12841,

25000000000 = 29 • 511,

25600000000 = 216 • 58,

25711938560 = 210 • 5 • 73 • 114,

25735718750 = 2 • 56 • 77,

27682574402 = 2 • 712,

28397229568 = 29 • 833 • 97,

31381059609 = 322,

35214691275 = 38 • 52 • 214691,

35595703125 = 36 • 511,

37822859361 = 38 • 78,

57382399375 = 54 • 74 • 38239,

57736239625 = 53 • 7733,

137858491849 = 1310,

231928233984 = 233 • 33,

232335362178 = 2 • 33 • 233 • 353621,

235092492288 = 214 • 315,

235737907200 = 216 • 33 • 52 • 732,

235929600000 = 223 • 32 • 55,

237916127232 = 219 • 33 • 75,

250000000000 = 210 • 512,

256000000000 = 217 • 59,

257119385600 = 211 • 52 • 73 • 114,

257357187500 = 22 • 57 • 77,

261993005056 = 232 • 61,

273162960896 = 219 • 75 • 31,

274877906944 = 238,

323754489243 = 37 • 236,

331353897741 = 35 • 31 • 3533,

351189498825 = 315 • 52 • 11 • 89,

351353851875 = 39 • 54 • 134,

353797811253 = 313 • 532 • 79,

361881028863 = 313 • 613,

373714754427 = 33 • 712,

511374199655 = 5 • 113 • 374 • 41,

557041015625 = 510 • 57041.

 

La tabella seguente mostra i numeri illuminati fino a 106, nelle basi fino a 20.

Base

Numeri illuminati

2

1002 = 4 = 22, 10002 = 8 = 23, 100002 = 16 = 24, 110112 = 27 = 33, 1000002 = 32 = 25, 1111112 = 63 = 32 • 7, 10000002 = 64 = 26, 100000002 = 128 = 27, 111100112 = 243 = 35, 1000000002 = 256 = 28, 1010111102 = 350 = 2 • 52 • 7, 10000000002 = 512 = 29, 10101111002 = 700 = 22 • 52 • 7, 10111011102 = 750 = 2 • 3 • 53, 100000000002 = 1024 = 210, 101011110002 = 1400 = 23 • 52 • 7, 101101100102 = 1458 = 2 • 36, 101110111002 = 1500 = 22 • 3 • 53, 101111101012 = 1525 = 52 • 61, 111010100112 = 1875 = 3 • 54, 1000000000002 = 2048 = 211, 1010111100002 = 2800 = 24 • 52 • 7, 1011011001002 = 2916 = 22 • 36, 1011101110002 = 3000 = 23 • 3 • 53, 1111011011112 = 3951 = 32 • 439, 1111100000012 = 3969 = 34 • 72, 10000000000002 = 4096 = 212, 10101111000002 = 5600 = 25 • 52 • 7, 10110110010002 = 5832 = 23 • 36, 10110111100112 = 5875 = 53 • 47, 10111011100002 = 6000 = 24 • 3 • 53, 10111111011012 = 6125 = 53 • 72, 11001101000012 = 6561 = 38, 100000000000002 = 8192 = 213, 101011110000002 = 11200 = 26 • 52 • 7, 101101100100002 = 11664 = 24 • 36, 101110111000002 = 12000 = 25 • 3 • 53, 101111100001112 = 12167 = 233, 1000000000000002 = 16384 = 214, 1010111100000002 = 22400 = 27 • 52 • 7, 1011011001000002 = 23328 = 25 • 36, 1011101100101102 = 23958 = 2 • 32 • 113, 1011101110000002 = 24000 = 26 • 3 • 53, 1101111100100012 = 28561 = 134, 1110100110000012 = 29889 = 36 • 41, 1110110101001112 = 30375 = 35 • 53, 1111011011011112 = 31599 = 32 • 3511, 10000000000000002 = 32768 = 215, 10101111000000002 = 44800 = 28 • 52 • 7, 10110110010000002 = 46656 = 26 • 36, 10111011001011002 = 47916 = 22 • 32 • 113, 10111011100000002 = 48000 = 27 • 3 • 53, 10111011111100102 = 48114 = 2 • 37 • 11, 11100110101010012 = 59049 = 310, 11111101001110112 = 64827 = 33 • 74, 100000000000000002 = 65536 = 216, 101011110000000002 = 89600 = 29 • 52 • 7, 101101100100000002 = 93312 = 27 • 36, 101110110010110002 = 95832 = 23 • 32 • 113, 101110111000000002 = 96000 = 28 • 3 • 53, 101110111111001002 = 96228 = 22 • 37 • 11, 111001011100100012 = 117649 = 76, 111011111101010012 = 122793 = 3 • 11 • 612, 1000000000000000002 = 131072 = 217, 1010001100100000102 = 167042 = 2 • 174, 1010111100000000002 = 179200 = 210 • 52 • 7, 1011011001000000002 = 186624 = 28 • 36, 1011101100101100002 = 191664 = 24 • 32 • 113, 1011101110000000002 = 192000 = 29 • 3 • 53, 1011101111110010002 = 192456 = 23 • 37 • 11, 1011110001101011102 = 192942 = 2 • 35 • 397, 1101110011111100112 = 226291 = 133 • 103, 1111011011011011112 = 252783 = 32 • 28087, 1111101011100010112 = 256907 = 74 • 107, 1111110010101101012 = 258741 = 33 • 7 • 372, 10000000000000000002 = 262144 = 218, 10100011001000001002 = 334084 = 22 • 174, 10101111000000000002 = 358400 = 211 • 52 • 7, 10110110010000000002 = 373248 = 29 • 36, 10111011001011000002 = 383328 = 25 • 32 • 113, 10111011100000000002 = 384000 = 210 • 3 • 53, 10111011111100100002 = 384912 = 24 • 37 • 11, 10111100011010111002 = 385884 = 22 • 35 • 397, 10111110101111000012 = 390625 = 58, 11111110101111001112 = 521703 = 32 • 73 • 132, 100000000000000000002 = 524288 = 219, 101000110010000010002 = 668168 = 23 • 174, 101011110000000000002 = 716800 = 212 • 52 • 7, 101101010100101110102 = 742586 = 2 • 135, 101101100100000000002 = 746496 = 210 • 36, 101110001110010110112 = 757339 = 113 • 569, 101110110010110000002 = 766656 = 26 • 32 • 113, 101110111000000000002 = 768000 = 211 • 3 • 53, 101110111111001000002 = 769824 = 25 • 37 • 11, 101111000110101110002 = 771768 = 23 • 35 • 397, 101111010111001001012 = 775973 = 114 • 53, 101111110111111101112 = 784375 = 55 • 251, 111011001110001110112 = 970299 = 36 • 113, 111011011001100111112 = 973215 = 37 • 5 • 89, 111100111110100101012 = 999061 = 72 • 20389

3

223 = 8 = 23, 1003 = 9 = 32, 10003 = 27 = 33, 21013 = 64 = 26, 100003 = 81 = 34, 210103 = 192 = 26 • 3, 1000003 = 243 = 35, 2002223 = 512 = 29, 2101003 = 576 = 26 • 32, 2102103 = 588 = 22 • 3 • 72, 2122013 = 640 = 27 • 5, 2211023 = 686 = 2 • 73, 10000003 = 729 = 36, 21010003 = 1728 = 26 • 33, 21021003 = 1764 = 22 • 32 • 72, 21110113 = 1813 = 72 • 37, 22102123 = 2048 = 211, 100000003 = 2187 = 37, 210100003 = 5184 = 26 • 34, 210210003 = 5292 = 22 • 33 • 72, 210220123 = 5324 = 22 • 113, 221210223 = 6272 = 27 • 72, 1000000003 = 6561 = 38, 1211111013 = 12025 = 52 • 13 • 37, 2101000003 = 15552 = 26 • 35, 2102100003 = 15876 = 22 • 34 • 72, 2111102113 = 16384 = 214, 2120011113 = 16807 = 75, 2122122023 = 17408 = 210 • 17, 10000000003 = 19683 = 39, 11100112113 = 28561 = 134, 21010000003 = 46656 = 26 • 36, 21021000003 = 47628 = 22 • 35 • 72, 21210200223 = 51200 = 211 • 52, 22021220213 = 54412 = 22 • 61 • 223, 100000000003 = 59049 = 310, 120211020223 = 103823 = 473, 122020112113 = 113125 = 54 • 181, 201212022013 = 130321 = 194, 201222101123 = 131072 = 217, 210100000003 = 139968 = 26 • 37, 210210000003 = 142884 = 22 • 36 • 72, 212211000013 = 156250 = 2 • 57, 212221222113 = 157216 = 25 • 173, 1000000000003 = 177147 = 311, 1012212100103 = 215625 = 3 • 55 • 23, 2101000000003 = 419904 = 26 • 38, 2102100000003 = 428652 = 22 • 37 • 72, 2201200000013 = 483328 = 213 • 59, 2221220120023 = 524288 = 219, 10000000000003 = 531441 = 312, 10122121001003 = 646875 = 32 • 55 • 23

4

204 = 8 = 23, 2004 = 32 = 25, 20004 = 128 = 27, 33034 = 243 = 35, 200004 = 512 = 29, 1133114 = 1525 = 52 • 61, 2000004 = 2048 = 211, 2133104 = 2548 = 22 • 72 • 13, 2312104 = 2916 = 22 • 36, 11233034 = 5875 = 53 • 47, 20000004 = 8192 = 213, 21331004 = 10192 = 24 • 72 • 13, 23121004 = 11664 = 24 • 36, 200000004 = 32768 = 215, 213310004 = 40768 = 26 • 72 • 13, 231210004 = 46656 = 26 • 36, 232302304 = 47916 = 22 • 32 • 113, 232333024 = 48114 = 2 • 37 • 11, 311121034 = 54675 = 37 • 52, 321222214 = 59049 = 310, 1302321014 = 117649 = 76, 2000000004 = 131072 = 217, 2133100004 = 163072 = 28 • 72 • 13, 2312100004 = 186624 = 28 • 36, 2323023004 = 191664 = 24 • 32 • 113, 2323330204 = 192456 = 23 • 37 • 11, 11331132014 = 390625 = 58, 20000000004 = 524288 = 219, 21331000004 = 652288 = 210 • 72 • 13, 23103122104 = 740772 = 22 • 33 • 193, 23111023224 = 742586 = 2 • 135, 23121000004 = 746496 = 210 • 36, 23203211234 = 757339 = 113 • 569, 23230230004 = 766656 = 26 • 32 • 113, 23233302004 = 769824 = 25 • 37 • 11, 23311302114 = 775973 = 114 • 53, 32303203234 = 970299 = 36 • 113

5

1005 = 25 = 52, 2245 = 64 = 26, 3115 = 81 = 34, 10005 = 125 = 53, 20115 = 256 = 28, 24335 = 368 = 24 • 23, 100005 = 625 = 54, 234035 = 1728 = 26 • 33, 234435 = 1748 = 22 • 19 • 23, 322225 = 2187 = 37, 1000005 = 3125 = 55, 2121335 = 7168 = 210 • 7, 2214345 = 7744 = 26 • 112, 2302325 = 8192 = 213, 2344435 = 8748 = 22 • 37, 10000005 = 15625 = 56, 20220335 = 32768 = 215, 23113425 = 41472 = 29 • 34, 31223425 = 51597 = 34 • 72 • 13, 33421445 = 59049 = 310, 100000005 = 78125 = 57, 122310445 = 117649 = 76, 210323105 = 174080 = 211 • 5 • 17, 232043025 = 209952 = 25 • 38, 232212315 = 210816 = 27 • 33 • 61, 1000000005 = 390625 = 58, 2103231005 = 870400 = 211 • 52 • 17, 2214131115 = 966656 = 214 • 59

6

246 = 16 = 24, 22126 = 512 = 29, 25446 = 640 = 27 • 5, 32136 = 729 = 36, 35136 = 837 = 33 • 31, 1155116 = 10339 = 72 • 211, 1514416 = 14641 = 114, 2035046 = 16384 = 214, 3523436 = 30375 = 35 • 53, 21010026 = 101306 = 2 • 373, 23104306 = 118098 = 2 • 310, 24504526 = 131072 = 217, 25154126 = 134720 = 26 • 5 • 421, 25215326 = 135200 = 25 • 52 • 132, 25253446 = 136000 = 26 • 53 • 17, 34440436 = 177147 = 311, 211013446 = 614656 = 28 • 74, 231043006 = 708588 = 22 • 311, 250514126 = 800000 = 28 • 55, 253203326 = 819200 = 215 • 52

7

227 = 16 = 24, 367 = 27 = 33, 1007 = 49 = 72, 2427 = 128 = 27, 10007 = 343 = 73, 23437 = 864 = 25 • 33, 25257 = 950 = 2 • 52 • 19, 26627 = 1024 = 210, 100007 = 2401 = 74, 225617 = 5776 = 24 • 192, 230017 = 5832 = 23 • 36, 236257 = 6144 = 211 • 3, 256667 = 6859 = 193, 1000007 = 16807 = 75, 2316247 = 41472 = 29 • 34, 2341627 = 42282 = 2 • 36 • 29, 3341047 = 59049 = 310, 10000007 = 117649 = 76, 16523447 = 231361 = 132 • 372, 21411617 = 262144 = 218, 23355427 = 294912 = 215 • 32, 23564317 = 300000 = 25 • 3 • 55, 25016427 = 320000 = 29 • 54, 25332237 = 327680 = 216 • 5, 25661357 = 335872 = 213 • 41, 35145637 = 441045 = 36 • 5 • 112, 100000007 = 823543 = 77

8

208 = 16 = 24, 338 = 27 = 33, 2008 = 128 = 27, 3638 = 243 = 35, 20008 = 1024 = 210, 25708 = 1400 = 23 • 52 • 7, 35238 = 1875 = 3 • 54, 132738 = 5819 = 11 • 232, 200008 = 8192 = 213, 213778 = 8959 = 172 • 31, 237368 = 10206 = 2 • 36 • 7, 257008 = 11200 = 26 • 52 • 7, 276078 = 12167 = 233, 2000008 = 65536 = 216, 2316148 = 78732 = 22 • 39, 2373608 = 81648 = 24 • 36 • 7, 2570008 = 89600 = 29 • 52 • 7, 3233238 = 108243 = 33 • 19 • 211, 20000008 = 524288 = 219, 23161408 = 629856 = 25 • 39, 23736008 = 653184 = 27 • 36 • 7, 25700008 = 716800 = 212 • 52 • 7

9

309 = 27 = 33, 2339 = 192 = 26 • 3, 3009 = 243 = 35, 23309 = 1728 = 26 • 33, 23709 = 1764 = 22 • 32 • 72, 27259 = 2048 = 211, 30009 = 2187 = 37, 233009 = 15552 = 26 • 35, 237009 = 15876 = 22 • 34 • 72, 244249 = 16384 = 214, 300009 = 19683 = 39, 2176819 = 130321 = 194, 2187159 = 131072 = 217, 2256819 = 135424 = 28 • 232, 2330009 = 139968 = 26 • 37, 2370009 = 142884 = 22 • 36 • 72, 2573019 = 156250 = 2 • 57, 3000009 = 177147 = 311, 5117259 = 303125 = 55 • 97

10

250 = 2 • 53, 256 = 28, 2048 = 211, 2176 = 27 • 17, 2304 = 28 • 32, 2500 = 22 • 54, 2560 = 29 • 5, 2744 = 23 • 73, 23328 = 25 • 36, 25000 = 23 • 55, 25600 = 210 • 52, 119911 = 112 • 991, 219488 = 25 • 193, 236196 = 22 • 310, 250000 = 24 • 56, 256000 = 211 • 53, 262144 = 218, 290912 = 25 • 9091

11

2A11 = 32 = 25, 10011 = 121 = 112, 21311 = 256 = 28, 37111 = 441 = 32 • 72, 51911 = 625 = 54, 100011 = 1331 = 113, 234311 = 3072 = 210 • 3, 276911 = 3584 = 29 • 7, 371011 = 4851 = 32 • 72 • 11, 1000011 = 14641 = 114, 2268A11 = 32768 = 215, 2321611 = 33534 = 2 • 36 • 23, 3504311 = 50625 = 34 • 54, 3710011 = 53361 = 32 • 72 • 112, 5377311 = 78125 = 57, 7436111 = 108241 = 72 • 472, 10000011 = 161051 = 115, 23547711 = 373248 = 29 • 36, 24330211 = 385024 = 213 • 47, 25358711 = 400000 = 27 • 55, 25A81411 = 409600 = 214 • 52, 33330911 = 531441 = 312, 37100011 = 586971 = 32 • 72 • 113, 53493111 = 855625 = 54 • 372

12

2812 = 32 = 25, 23012 = 324 = 22 • 34, 230012 = 3888 = 24 • 35, 245412 = 4096 = 212, 396912 = 6561 = 38, 2300012 = 46656 = 26 • 36, 2576812 = 51200 = 211 • 52, 2959412 = 57856 = 29 • 113, 2BA2812 = 61952 = 29 • 112, 5116112 = 105625 = 54 • 132, 2134A812 = 524288 = 219, 23000012 = 559872 = 28 • 37

13

2613 = 32 = 25, 10013 = 169 = 132, 23713 = 384 = 27 • 3, 2B313 = 484 = 22 • 112, 100013 = 2197 = 133, 214913 = 4624 = 24 • 172, 376513 = 7857 = 34 • 97, 785B13 = 16807 = 75, 1000013 = 28561 = 134, 23AA313 = 65536 = 216, 252C713 = 68608 = 210 • 67, 2CBC313 = 85504 = 29 • 167, 35A4513 = 98415 = 39 • 5, 3834513 = 103823 = 473, 10000013 = 371293 = 135, 23533813 = 839808 = 27 • 38

14

2414 = 32 = 25, 28814 = 512 = 29, 3A114 = 729 = 36, 234C14 = 6144 = 211 • 3, 237014 = 6174 = 2 • 32 • 73, 2B7214 = 7744 = 26 • 112, 2DB214 = 8192 = 213, 35B914 = 9375 = 3 • 55, 59A114 = 15625 = 56, 2370014 = 86436 = 22 • 32 • 74, 2D8C414 = 114244 = 22 • 134, 35D7B14 = 131625 = 34 • 53 • 13

15

2215 = 32 = 25, 24215 = 512 = 29, 33915 = 729 = 36, 22D115 = 7396 = 22 • 432, 235015 = 7500 = 22 • 3 • 54, 258515 = 8000 = 26 • 53, 266215 = 8192 = 213, 37DC15 = 11907 = 35 • 72, 2350015 = 112500 = 22 • 32 • 55, 23BD315 = 114048 = 27 • 34 • 11, 287ED15 = 130048 = 210 • 127, 28C8215 = 131072 = 217, 3774C15 = 177147 = 311

16

2016 = 32 = 25, 20016 = 512 = 29, 200016 = 8192 = 213, 237016 = 9072 = 24 • 34 • 7, 27D016 = 10192 = 24 • 72 • 13, 2000016 = 131072 = 217, 2370016 = 145152 = 28 • 34 • 7, 27D0016 = 163072 = 28 • 72 • 13, 5F5E116 = 390625 = 58

17

10017 = 289 = 172, 100017 = 4913 = 173, 345717 = 15987 = 3 • 732, 1000017 = 83521 = 174, 23GCF17 = 186624 = 28 • 36, B29A917 = 931337 = 112 • 43 • 179

18

27A18 = 784 = 24 • 72, 77718 = 2401 = 74, 25D218 = 13520 = 24 • 5 • 132, 25HA18 = 13600 = 25 • 52 • 17, 2BE818 = 15488 = 27 • 112, 2EA418 = 16384 = 214, 2HD218 = 17408 = 210 • 17, 36D918 = 19683 = 39, 5D5D18 = 33475 = 52 • 13 • 103, 28H1A18 = 262144 = 218, 2BH7E18 = 279752 = 23 • 112 • 172, 2HG8818 = 314432 = 26 • 173, 3F7D918 = 404919 = 34 • 4999, 7F3E718 = 823543 = 77

19

10019 = 361 = 192, 2FH19 = 1024 = 210, 100019 = 6859 = 193, 234719 = 14884 = 22 • 612, 277619 = 16384 = 214, 31AD19 = 21141 = 36 • 29, 1000019 = 130321 = 194, 2043119 = 262144 = 218, 223CD19 = 275684 = 22 • 413, 2BCG919 = 340736 = 28 • 113, 2E5H819 = 358804 = 22 • 271 • 331, 2F47D19 = 365117 = 53 • 832, 476H119 = 571787 = 833

20

23420 = 864 = 25 • 33, 2B420 = 1024 = 210, 2HG20 = 1156 = 22 • 172, 20J420 = 16384 = 214, 23B420 = 17424 = 24 • 32 • 112, 23EG20 = 17496 = 23 • 37, 573F20 = 42875 = 53 • 73, 2783420 = 379264 = 27 • 2963, 2D4J420 = 425984 = 215 • 13, 368C120 = 531441 = 312

 

Esistono numeri illuminati in più basi; per lo più sono potenze di numeri primi, per esempio, 134217728 = 227 è illuminato in base 2, 4, 5, 6, 13, 19 20, ma vi sono anche rari altri esempi, come 1728 = 26 • 33, illuminato in base 3, 5 e 9.

 

I numeri illuminati noti che non siano un quadrato di un primo o multipli di un cubo sono relativamente rari; la tabella seguente riporta quelli fino a 109 nelle basi fino a 20.

Base

Numeri illuminati

2

1111112 = 63 = 32 • 7, 1010111102 = 350 = 2 • 52 • 7, 10101111002 = 700 = 22 • 52 • 7, 101111101012 = 1525 = 52 • 61, 1111011011011112 = 31599 = 32 • 3511, 111011111101010012 = 122793 = 3 • 11 • 612, 1111011011011011112 = 252783 = 32 • 28087, 111100111110100101012 = 999061 = 72 • 20389, 101001111000111011001012 = 5490533 = 832 • 797, 1110001100000111011110112 = 14878587 = 3 • 172 • 1312, 10110000110100111010101012 = 23177045 = 5 • 21532, 11011010100011111101111112 = 28647359 = 132 • 337 • 503, 1010011111000111111110010012 = 87965641 = 832 • 1132, 1011111101001110101110000012 = 100300225 = 52 • 20032, 1110001100111001110110111012 = 119131869 = 3 • 172 • 313 • 439, 1110111010000011100100101012 = 125050005 = 32 • 5 • 16672, 111111100111111010011011000012 = 533713761 = 32 • 7 • 103 • 233 • 353, 1001111111010011110011011101012 = 670364533 = 792 • 233 • 461, 1011111010011111100010111111112 = 799531775 = 52 • 72 • 41 • 15919

3

2102103 = 588 = 22 • 3 • 72, 21021003 = 1764 = 22 • 32 • 72, 21110113 = 1813 = 72 • 37, 1111011011112 = 3951 = 32 • 439, 1211111013 = 12025 = 52 • 13 • 37, 22021220213 = 54412 = 22 • 61 • 223, 102221110120203 = 2115699 = 3 • 792 • 113, 1022211101202003 = 6347097 = 32 • 792 • 113

4

1133114 = 1525 = 52 • 61, 2133104 = 2548 = 22 • 72 • 13, 1103301312114 = 5490533 = 832 • 797, 11201221311114 = 23177045 = 5 • 21532

5

234435 = 1748 = 22 • 19 • 23, 23333344242435 = 671404948 = 22 • 11719 • 14323

6

1155116 = 10339 = 72 • 211, 255110415246 = 180051100 = 22 • 52 • 31 • 2412

7

25257 = 950 = 2 • 52 • 19, 16523447 = 231361 = 132 • 372

8

132738 = 5819 = 11 • 232, 213778 = 8959 = 172 • 31, 1552176778 = 28647359 = 132 • 337 • 503

9

23709 = 1764 = 22 • 32 • 72

10

119911 = 112 • 991

11

37111 = 441 = 32 • 72, 371011 = 4851 = 32 • 72 • 11, 3710011 = 53361 = 32 • 72 • 112, 7436111 = 108241 = 72 • 472, 23776515A11 = 500802246 = 2 • 32 • 72 • 83 • 6841

12

B1072112 = 2758921 = 112 • 1512

13

2B313 = 484 = 22 • 112

14

2954D314 = 1436081 = 372 • 1049

15

22D115 = 7396 = 22 • 432

16

D1DBF916 = 13753337 = 13 • 29 • 1912

17

345717 = 15987 = 3 • 732, B29A917 = 931337 = 112 • 43 • 179

18

5D5D18 = 33475 = 52 • 13 • 103, BD726518 = 22191521 = 112 • 241 • 761

19

234719 = 14884 = 22 • 612, 2E5H819 = 358804 = 22 • 271 • 331, 2F47D19 = 365117 = 53 • 832

20

2HG20 = 1156 = 22 • 172

 

In base 10 non ne esistono altri minori di 1012( Giovanni Resta).

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