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Reciprocamente descrittivi (numeri) (II)

Rappresentazione dei numeri 

Si possono definire “numeri reciprocamente descrittivi” o “reciprocamente biografici” i numeri naturali che formano coppie nelle quali ciascuno “descrive” l’altro, nel senso dei numeri autodescrittivi (II). Vale a dire che concatenando i numeri di cifre che compongono uno dei numeri, in ordine crescente a partire da 0, si ottiene l’altro numero.

Per esempio, nella minima coppia, 130 e 1101, il primo descrive il secondo come formato da uno 0, tre 1 e nessun 2, il secondo descrive il primo come formato da uno 0, un 1, nessun 2 e un 3.

Dato che possono essere formati al massimo da un numero di cifre uguale alla base, sono in numero finito in qualsiasi base.

 

Come per i numeri autodescrittivi (II), una coppia di sequenze di cifre reciprocamente descrittive in una base resta tale in tutte le basi superiori, anche se rappresenta numeri diversi; dato che la prima cifra deve essere almeno 1, devono contenere almeno uno zero.

 

Non esistono numeri reciprocamente descrittivi nelle basi 2 e 3.

 

Si può allargare la definizione, ammettendo numeri con più cifre della base; in questo case le cifre corrispondenti ai posti dalla base in poi sono per forza zero e e devono essere in totale meno della base stessa, perché il loro numero non sarebbe rappresentabile, quindi le cifre sono al massimo il doppio della base meno uno e il numero di coppie resta finito.

Per esempio, in base 10 possiamo avere numeri con zeri dall’undicesima posizione in poi, perché non possono esserci cifre maggiori di 9, e al massimo 9 zeri, perché altrimenti il numero di zeri non sarebbe rappresentabile in base 10, quindi 19 cifre in tutto.

 

Con la definizione allargata vi sono 86 coppie in base 10: (130, 1101), (230, 10110), (430, 1001100), (530, 10010100), (630, 100100100), (730, 1001000100), (830, 10010000100), (930, 100100000100), (2210, 11200), (23100, 211100), (43100, 21011000), (53100, 210101000), (63100, 2101001000), (73100, 21010001000), (83100, 210100001000), (93100, 2101000001000), (230100, 311100), (411100, 2300100), (430100, 31011000), (511100, 23000100), (530100, 310101000), (611100, 230000100), (630100, 3101001000), (711100, 2300000100), (730100, 31010001000), (811100, 23000000100), (830100, 310100001000), (911100, 230000000100), (930100, 3101000001000), (4211000, 32101000), (4300100, 41011000), (5211000, 321001000), (5300100, 410101000), (6211000, 3210001000), (6300100, 4101001000), (7211000, 32100001000), (7300100, 41010001000), (8211000, 321000001000), (8300100, 410100001000), (9211000, 3210000001000), (9300100, 4101000001000), (43000100, 51011000), (52101000, 421001000), (53000100, 510101000), (61011000, 430000100), (62101000, 4210001000), (63000100, 5101001000), (71011000, 4300000100), (72101000, 42100001000), (73000100, 51010001000), (81011000, 43000000100), (82101000, 421000001000), (83000100, 510100001000), (91011000, 430000000100), (92101000, 4210000001000), (93000100, 5101000001000), (530000100, 610101000), (621001000, 5210001000), (630000100, 6101001000), (710101000, 5300000100), (721001000, 52100001000), (730000100, 61010001000), (810101000, 53000000100), (821001000, 521000001000), (830000100, 610100001000), (910101000, 530000000100), (921001000, 5210000001000), (930000100, 6101000001000), (6300000100, 7101001000), (7210001000, 62100001000), (7300000100, 71010001000), (8101001000, 63000000100), (8210001000, 621000001000),(8300000100, 710100001000), (9101001000, 630000000100), (9210001000, 6210000001000), (9300000100, 7101000001000), (73000000100, 81010001000), (83000000100, 810100001000), (91010001000, 730000000100), (92100001000, 7210000001000), (93000000100, 8101000001000), (721000001000, 82100001000), (830000000100, 910100001000), (921000001000, 8210000001000), (930000000100, 9101000001000) (Martin Fuller, The Online Encyclopedia of Integer Sequences, http://oeis.org).

 

La tabella seguente riporta le coppie di numeri reciprocamente descrittivi, ammettendo numeri con più cifre della base, col numero minore della coppia non superiore a 106, nelle basi fino a 20.

Base

Numeri reciprocamente descrittivi

4

(1304 = 28, 11014 = 81), (2304 = 44, 101104 = 276), (22104 = 164, 112004 = 352)

5

(1305 = 40, 11015 = 151), (2305 = 65, 101105 = 655), (4305 = 115, 10011005 = 15775), (22105 = 305, 112005 = 800), (231005 = 1650, 2111005 = 7025), (431005 = 2900, 210110005 = 172625)

6

(1306 = 54, 11016 = 253), (2306 = 90, 101106 = 1338), (4306 = 162, 10011006 = 46908), (5306 = 198, 100101006 = 281268), (22106 = 510, 112006 = 1584), (231006 = 3276, 2111006 = 17100), (431006 = 5868, 210110006 = 608040), (531006 = 7164, 2101010006 = 3647160), (2301006 = 19476, 3111006 = 24876), (4111006 = 32652, 23001006 = 116676), (4301006 = 35028, 310110006 = 887976), (5111006 = 40428, 230001006 = 699876), (5301006 = 42804, 3101010006 = 5326776)

7

(1307 = 70, 11017 = 393), (2307 = 119, 101107 = 2457), (4307 = 217, 10011007 = 118041), (5307 = 266, 100101007 = 825993), (6307 = 315, 1001001007 = 5781657), (22107 = 791, 112007 = 2842), (231007 = 5880, 2111007 = 36407), (431007 = 10682, 210110007 = 1767479), (531007 = 13083, 2101010007 = 12370295), (631007 = 15484, 21010010007 = 86590007), (2301007 = 40866, 3111007 = 53214), (4111007 = 70021, 23001007 = 285768), (4301007 = 74480, 310110007 = 2591022), (5111007 = 86828, 230001007 = 2000082), (5301007 = 91287, 3101010007 = 18135096), (6111007 = 103635, 2300001007 = 14000280), (6301007 = 108094, 31010010007 = 126943614), (42110007 = 506954, 321010007 = 2723077), (43001007 = 521066, 410110007 = 3414565), (52110007 = 624603, 3210010007 = 19059481), (53001007 = 638715, 4101010007 = 23899897), (62110007 = 742252, 32100010007 = 133414309), (63001007 = 756364, 41010010007 = 167297221)

8

(1308 = 88, 11018 = 577), (2308 = 152, 101108 = 4168), (4308 = 280, 10011008 = 262720), (5308 = 344, 100101008 = 2101312), (6308 = 408, 1001001008 = 16810048), (7308 = 472, 10010001008 = 134479936), (22108 = 1160, 112008 = 4736), (231008 = 9792, 2111008 = 70208), (431008 = 17984, 210110008 = 4461056), (531008 = 22080, 2101010008 = 35684864), (631008 = 26176, 21010010008 = 285475328), (731008 = 30272, 210100010008 = 2283799040), (2301008 = 77888, 3111008 = 102976), (4111008 = 135744, 23001008 = 622656), (4301008 = 143424, 310110008 = 6558208), (5111008 = 168512, 230001008 = 4980800), (5301008 = 176192, 3101010008 = 52462080), (6111008 = 201280, 2300001008 = 39845952), (6301008 = 208960, 31010010008 = 419693056), (7111008 = 234048, 23000001008 = 318767168), (7301008 = 241728, 310100010008 = 3357540864)

9

(1309 = 108, 11019 = 811), (2309 = 189, 101109 = 6651), (4309 = 351, 10011009 = 532251), (5309 = 432, 100101009 = 4789611), (6309 = 513, 1001001009 = 43105851), (7309 = 594, 10010001009 = 387952011), (8309 = 675, 100100001009 = 3491567451), (22109 = 1629, 112009 = 7452), (231009 = 15390, 2111009 = 125469), (431009 = 28512, 210110009 = 10104669), (531009 = 35073, 2101010009 = 90936189), (631009 = 41634, 21010010009 = 818419869), (731009 = 48195, 210100010009 = 7365772989), (831009 = 54756, 2101000010009 = 66291951069), (2301009 = 137862, 3111009 = 184518), (4111009 = 243567, 23001009 = 1240110), (4301009 = 255960, 310110009 = 14887638), (5111009 = 302616, 230001009 = 11160342), (5301009 = 315009, 3101010009 = 133982910), (6111009 = 361665, 2300001009 = 100442430), (6301009 = 374058, 31010010009 = 1205840358), (7111009 = 420714, 23000001009 = 903981222), (7301009 = 433107, 310100010009 = 10852557390), (8111009 = 479763, 230000001009 = 8135830350), (8301009 = 492156, 3101000010009 = 97673010678)

10

(130, 1101), (230, 10110), (430, 1001100), (530, 10010100), (630, 100100100), (730, 1001000100), (830, 10010000100), (930, 100100000100), (2210, 11200), (23100, 211100), (43100, 21011000), (53100, 210101000), (63100, 2101001000), (73100, 21010001000), (83100, 210100001000), (93100, 2101000001000), (230100, 311100), (411100, 2300100), (430100, 31011000), (511100, 23000100), (530100, 310101000), (611100, 230000100), (630100, 3101001000), (711100, 2300000100), (730100, 31010001000), (811100, 23000000100), (830100, 310100001000), (911100, 230000000100), (930100, 3101000001000)

11

(13011 = 154, 110111 = 1453), (23011 = 275, 1011011 = 14773), (43011 = 517, 100110011 = 1773013), (53011 = 638, 1001010011 = 19501933), (63011 = 759, 10010010011 = 214520053), (73011 = 880, 100100010011 = 2359719373), (83011 = 1001, 1001000010011 = 25956911893), (93011 = 1122, 10010000010011 = 285526029613), (A3011 = 1243, 100100000010011 = 3140786324533), (221011 = 2915, 1120011 = 16214), (2310011 = 33396, 21110011 = 338195), (4310011 = 62678, 2101100011 = 40761875), (5310011 = 77319, 21010100011 = 448367315), (6310011 = 91960, 210100100011 = 4932027155), (7310011 = 106601, 2101000100011 = 54252285395), (8310011 = 121242, 21010000100011 = 596775126035), (9310011 = 135883, 210100000100011 = 6564526373075), (A310011 = 150524, 2101000000100011 = 72209790090515), (23010011 = 366146, 31110011 = 499246), (41110011 = 660297, 230010011 = 4026396), (43010011 = 688248, 3101100011 = 60249046), (51110011 = 821348, 2300010011 = 44289146), (53010011 = 849299, 31010100011 = 662726196), (61110011 = 982399, 23000010011 = 487179396)

12

(13012 = 180, 110112 = 1873), (23012 = 324, 1011012 = 20892), (43012 = 612, 100110012 = 2987856), (53012 = 756, 1001010012 = 35852688), (63012 = 900, 10010010012 = 430230672), (73012 = 1044, 100100010012 = 5162766480), (83012 = 1188, 1001000010012 = 61953196176), (93012 = 1332, 10010000010012 = 743438352528), (A3012 = 1476, 100100000010012 = 8921260228752), (B3012 = 1620, 1001000000010012 = 107055122743440), (221012 = 3756, 1120012 = 22752), (2310012 = 46800, 21110012 = 520272), (4310012 = 88272, 2101100012 = 74672064), (5310012 = 109008, 21010100012 = 896045760), (6310012 = 129744, 210100100012 = 10752530112), (7310012 = 150480, 2101000100012 = 129030342336), (8310012 = 171216, 21010000100012 = 1548364089024), (9310012 = 191952, 210100000100012 = 18580369049280), (A310012 = 212688, 2101000000100012 = 222964428572352), (B310012 = 233424, 21010000000100012 = 2675573142849216), (23010012 = 560016, 31110012 = 769104)

13

(13013 = 208, 110113 = 2367), (23013 = 377, 1011013 = 28743), (43013 = 715, 100110013 = 4829175), (53013 = 884, 1001010013 = 62777247), (63013 = 1053, 10010010013 = 816102183), (73013 = 1222, 100100010013 = 10609326351), (83013 = 1391, 1001000010013 = 137921240535), (93013 = 1560, 10010000010013 = 1792976124927), (A3013 = 1729, 100100000010013 = 23308689622023), (B3013 = 1898, 1001000000010013 = 303012965084271), (C3013 = 2067, 10010000000010013 = 3939168546093495), (221013 = 4745, 1120013 = 31096), (2310013 = 63882, 21110013 = 773513), (4310013 = 121004, 2101100013 = 130354601), (5310013 = 149565, 21010100013 = 1694583449), (6310013 = 178126, 210100100013 = 22029558473), (7310013 = 206687, 2101000100013 = 286384233785), (8310013 = 235248, 21010000100013 = 3722995012841), (9310013 = 263809, 210100000100013 = 48398935140569), (A310013 = 292370, 2101000000100013 = 629186156801033), (B310013 = 320931, 21010000000100013 = 8179420038387065), (C310013 = 349492, 210100000000100013 = 106332460499005481), (23010013 = 828438, 31110013 = 1144806)

14

(13014 = 238, 110114 = 2941), (23014 = 434, 1011014 = 38626), (43014 = 826, 100110014 = 7532476), (53014 = 1022, 1001010014 = 105452116), (63014 = 1218, 10010010014 = 1476327076), (73014 = 1414, 100100010014 = 20668576516), (83014 = 1610, 1001000010014 = 289360068676), (93014 = 1806, 10010000010014 = 4051040958916), (A3014 = 2002, 100100000010014 = 56714573422276), (B3014 = 2198, 1001000000010014 = 794004027909316), (C3014 = 2394, 10010000000010014 = 11116056390727876), (D3014 = 2590, 100100000000010014 = 155624789470187716), (221014 = 5894, 1120014 = 41552), (2310014 = 85260, 21110014 = 1117004), (4310014 = 162092, 2101100014 = 218397704), (5310014 = 200508, 21010100014 = 3057532184), (6310014 = 238924, 210100100014 = 42805414904), (7310014 = 277340, 2101000100014 = 599275772984), (8310014 = 315756, 21010000100014 = 8389860786104), (9310014 = 354172, 210100000100014 = 117458050969784), (A310014 = 392588, 2101000000100014 = 1644412713541304), (B310014 = 431004, 21010000000100014 = 23021777989542584), (C310014 = 469420, 210100000000100014 = 322304891853560504), (D310014 = 507836, 2101000000000100014 = 4512268485949811384)

15

(13015 = 270, 110115 = 3601), (23015 = 495, 1011015 = 50865), (43015 = 945, 100110015 = 11394225), (53015 = 1170, 1001010015 = 170910225), (63015 = 1395, 10010010015 = 2563650225), (73015 = 1620, 100100010015 = 38454750225), (83015 = 1845, 1001000010015 = 576821250225), (93015 = 2070, 10010000010015 = 8652318750225), (A3015 = 2295, 100100000010015 = 129784781250225), (B3015 = 2520, 1001000000010015 = 1946771718750225), (C3015 = 2745, 10010000000010015 = 29201575781250225), (D3015 = 2970, 100100000000010015 = 438023636718750225), (E3015 = 3195, 1001000000000010015 = 6570354550781250225), (221015 = 7215, 1120015 = 54450), (2310015 = 111600, 21110015 = 1572975), (4310015 = 212850, 2101100015 = 353163375), (5310015 = 263475, 21010100015 = 5297403375), (6310015 = 314100, 210100100015 = 79461003375), (7310015 = 364725, 2101000100015 = 1191915003375), (8310015 = 415350, 21010000100015 = 17878725003375), (9310015 = 465975, 210100000100015 = 268180875003375), (A310015 = 516600, 2101000000100015 = 4022713125003375), (B310015 = 567225, 21010000000100015 = 60340696875003375), (C310015 = 617850, 210100000000100015 = 905110453125003375), (D310015 = 668475, 2101000000000100015 = 13576656796875003375), (E310015 = 719100, 21010000000000100015 = 203649851953125003375)

16

(13016 = 304, 110116 = 4353), (23016 = 560, 1011016 = 65808), (43016 = 1072, 100110016 = 16781568), (53016 = 1328, 1001010016 = 268501248), (63016 = 1584, 10010010016 = 4296016128), (73016 = 1840, 100100010016 = 68736254208), (83016 = 2096, 1001000010016 = 1099780063488), (93016 = 2352, 10010000010016 = 17596481011968), (A3016 = 2608, 100100000010016 = 281543696187648), (B3016 = 2864, 1001000000010016 = 4504699138998528), (C3016 = 3120, 10010000000010016 = 72075186223972608), (D3016 = 3376, 100100000000010016 = 1153202979583557888), (E3016 = 3632, 1001000000000010016 = 18451247673336922368), (F3016 = 3888, 10010000000000010016 = 295219962773390754048), (221016 = 8720, 1120016 = 70144), (2310016 = 143616, 21110016 = 2167040), (4310016 = 274688, 2101100016 = 553717760), (5310016 = 340224, 21010100016 = 8859422720), (6310016 = 405760, 210100100016 = 141750702080), (7310016 = 471296, 2101000100016 = 2268011171840), (8310016 = 536832, 21010000100016 = 36288178688000), (9310016 = 602368, 210100000100016 = 580610858946560), (A310016 = 667904, 2101000000100016 = 9289773743083520), (B310016 = 733440, 21010000000100016 = 148636379889274880), (C310016 = 798976, 210100000000100016 = 2378182078228336640), (D310016 = 864512, 2101000000000100016 = 38050913251653324800), (E310016 = 930048, 21010000000000100016 = 608814612026453135360), (F310016 = 995584, 210100000000000100016 = 9741033792423250104320)

17

(13017 = 340, 110117 = 5203), (23017 = 629, 1011017 = 83827), (43017 = 1207, 100110017 = 24142771), (53017 = 1496, 1001010017 = 410422483), (63017 = 1785, 10010010017 = 6977177587), (73017 = 2074, 100100010017 = 118612014355), (83017 = 2363, 1001000010017 = 2016404239411), (93017 = 2652, 10010000010017 = 34278872065363), (A3017 = 2941, 100100000010017 = 582740825106547), (B3017 = 3230, 1001000000010017 = 9906594026806675), (C3017 = 3519, 10010000000010017 = 168412098455708851), (D3017 = 3808, 100100000000010017 = 2863005673747045843), (E3017 = 4097, 1001000000000010017 = 48671096453699774707), (F3017 = 4386, 10010000000000010017 = 827408639712896165395), (G3017 = 4675, 100100000000000010017 = 14065946875119234807091), (221017 = 10421, 1120017 = 89012), (2310017 = 182070, 21110017 = 2928437), (4310017 = 349112, 2101100017 = 844903349), (5310017 = 432633, 21010100017 = 14363278325), (6310017 = 516154, 210100100017 = 244175652917), (7310017 = 599675, 2101000100017 = 4150986020981), (8310017 = 683196, 21010000100017 = 70566762278069), (9310017 = 766717, 210100000100017 = 1199634958648565), (A310017 = 850238, 2101000000100017 = 20393794296946997), (B310017 = 933759, 21010000000100017 = 346694503048020341)

18

(13018 = 378, 110118 = 6157), (23018 = 702, 1011018 = 105318), (43018 = 1350, 100110018 = 34018380), (53018 = 1674, 1001010018 = 612325332), (63018 = 1998, 10010010018 = 11021850468), (73018 = 2322, 100100010018 = 198393302916), (83018 = 2646, 1001000010018 = 3571079446980), (93018 = 2970, 10010000010018 = 64279430040132), (A3018 = 3294, 100100000010018 = 1157029740716868), (B3018 = 3618, 1001000000010018 = 20826535332898116), (C3018 = 3942, 10010000000010018 = 374877635992160580), (D3018 = 4266, 100100000000010018 = 6747797447858884932), (E3018 = 4590, 1001000000000010018 = 121460354061459923268), (F3018 = 4914, 10010000000000010018 = 2186286373106278613316), (G3018 = 5238, 100100000000000010018 = 39353154715913015034180), (H3018 = 5562, 1001000000000000010018 = 708356784886434270609732), (221018 = 12330, 1120018 = 111456), (2310018 = 227772, 21110018 = 3890268), (4310018 = 437724, 2101100018 = 1258563096), (5310018 = 542700, 21010100018 = 22654036584), (6310018 = 647676, 210100100018 = 407772559368), (7310018 = 752652, 2101000100018 = 7339905969480), (8310018 = 857628, 21010000100018 = 132118307351496), (9310018 = 962604, 210100000100018 = 2378129532227784),

19

(13019 = 418, 110119 = 7221), (23019 = 779, 1011019 = 130701), (43019 = 1501, 100110019 = 47053101), (53019 = 1862, 1001010019 = 894002421), (63019 = 2223, 10010010019 = 16986039501), (73019 = 2584, 100100010019 = 322734744021), (83019 = 2945, 1001000010019 = 6131960129901), (93019 = 3306, 10010000010019 = 116507242461621), (A3019 = 3667, 100100000010019 = 2213637606764301), (B3019 = 4028, 1001000000010019 = 42059114528515221), (C3019 = 4389, 10010000000010019 = 799123176041782701), (D3019 = 4750, 100100000000010019 = 15183340344793864821), (E3019 = 5111, 1001000000000010019 = 288483466551083425101), (F3019 = 5472, 10010000000000010019 = 5481185864470585070421), (G3019 = 5833, 100100000000000010019 = 104142531424941116331501), (H3019 = 6194, 1001000000000000010019 = 1978708097073881210292021), (I3019 = 6555, 10010000000000000010019 = 37595453844403742995541901), (221019 = 14459, 1120019 = 137902), (2310019 = 281580, 21110019 = 5089739), (4310019 = 542222, 2101100019 = 1834926539), (5310019 = 672543, 21010100019 = 34863480779), (6310019 = 802864, 210100100019 = 662406011339), (7310019 = 933185, 2101000100019 = 12585714091979)

20

(13020 = 460, 110120 = 8401), (23020 = 860, 1011020 = 160420), (43020 = 1660, 100110020 = 64008400), (53020 = 2060, 1001010020 = 1280160400), (63020 = 2460, 10010010020 = 25603200400), (73020 = 2860, 100100010020 = 512064000400), (83020 = 3260, 1001000010020 = 10241280000400), (93020 = 3660, 10010000010020 = 204825600000400), (A3020 = 4060, 100100000010020 = 4096512000000400), (B3020 = 4460, 1001000000010020 = 81930240000000400), (C3020 = 4860, 10010000000010020 = 1638604800000000400), (D3020 = 5260, 100100000000010020 = 32772096000000000400), (E3020 = 5660, 1001000000000010020 = 655441920000000000400), (F3020 = 6060, 10010000000000010020 = 13108838400000000000400), (G3020 = 6460, 100100000000000010020 = 262176768000000000000400), (H3020 = 6860, 1001000000000000010020 = 5243535360000000000000400), (I3020 = 7260, 10010000000000000010020 = 104870707200000000000000400), (J3020 = 7660, 100100000000000000010020 = 2097414144000000000000000400), (221020 = 16820, 1120020 = 168800), (2310020 = 344400, 21110020 = 6568400), (4310020 = 664400, 2101100020 = 2624168000), (5310020 = 824400, 21010100020 = 52483208000), (6310020 = 984400, 210100100020 = 1049664008000)

 

Tanya Khovanova dimostrò nel 2008 che se si parte da un intero qualsiasi e lo si descrive, nel senso spiegato sopra, eliminando però gli zeri in coda, iterando il procedimento si finisce in uno dei due cicli { 12, 011 } o { 22, 002, 201, 111, 03, 1001 }. Se invece si considerano descrizioni di esattamente 10 cifre, con zeri nel posto corrispondente alle cifre mancanti, si arriva sempre al ciclo { 6300000100, 7101001000 }.

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