I cicli nei quali terminano i numeri fino a 10 cifre, nelle basi sino a 20

Le tabelle seguenti riportano i cicli nei quali terminano i numeri fino a 10 cifre, nelle basi sino a 20; di ogni ciclo sono riportati il numero minimo (che può anche avere un numero di cifre inferiore) e, tra parentesi, la lunghezza del ciclo.

 

Cifre \ base

2

3

4

5

1

02 = 0 (1)

03 = 0 (1)

04 = 0 (1)

05 = 0 (1)

2

02 = 0 (1), 12 = 1 (1)

03 = 0 (1)

04 = 0 (1), 34 = 3 (2)

05 = 0 (1), 135 = 8 (1)

3

02 = 0 (1), 112 = 3 (1)

03 = 0 (1), 223 = 8 (2)

04 = 0 (1), 1324 = 30 (1)

05 = 0 (1), 1435 = 48 (2)

4

02 = 0 (1), 1112 = 7 (1), 10012 = 9 (1)

03 = 0 (1), 10123 = 32 (2)

04 = 0 (1), 13324 = 126 (2), 30214 = 201 (1)

05 = 0 (1), 30325 = 392 (1)

5

02 = 0 (1), 11112 = 15 (1), 101012 = 21 (1)

03 = 0 (1), 202113 = 184 (1)

04 = 0 (1), 233314 = 570 (2)

05 = 0 (1), 304325 = 1992 (4)

6

02 = 0 (1), 111112 = 31 (1), 1011012 = 45 (1), 1100012 = 49 (1)

03 = 0 (1), 1022123 = 320 (3)

04 = 0 (1), 2133124 = 2550 (1), 3102214 = 3369 (1), 3302014 = 3873 (1)

05 = 0 (1), 2144235 = 7488 (5)

7

02 = 0 (1), 1111112 = 63 (1), 10111012 = 93 (1), 11010012 = 105 (1)

03 = 0 (1), 20222113 = 1696 (2), 22021013 = 2008 (1)

04 = 0 (1), 32032114 = 14565 (1)

05 = 0 (1), 32043225 = 53712 (4)

8

02 = 0 (1), 11111112 = 127 (1), 101111012 = 189 (1), 110110012 = 217 (1), 111000012 = 225 (1)

03 = 0 (1), 210221113 = 5332 (1)

04 = 0 (1), 220332124 = 41958 (3), 311022214 = 54441 (1), 331022014 = 62625 (1), 333020014 = 64641 (1)

05 = 0 (1), 304444325 = 249992 (6)

9

02 = 0 (1), 111111112 = 255 (1), 1011111012 = 381 (1), 1101110012 = 441 (1), 1110100012 = 465 (1)

03 = 0 (1), 2022222113 = 15304 (3), 2202221013 = 18208 (2), 2220210013 = 19144 (1)

04 = 0 (1), 2213331124 = 171990 (1), 3210322114 = 234405 (1), 3320321014 = 254865 (1)

05 = 0 (1), 4320432115 = 1831056 (1)

10

02 = 0 (1), 1111111112 = 511 (1), 10111111012 = 765 (1), 11011110012 = 889 (1), 11101100012 = 945 (1), 11110000012 = 961 (1)

03 = 0 (1), 10222222123 = 26240 (4), 22102211013 = 55360 (1)

04 = 0 (1), 31110222214 = 873129 (1), 32203321114 = 954261 (1), 33110222014 = 1004193 (1), 33310220014 = 1036929 (1), 33330200014 = 1044993 (1)

05 = 0 (1), 30444444325 = 6249992 (4)

 

Cifre \ base

6

7

8

9

1

06 = 0 (1)

07 = 0 (1)

08 = 0 (1)

09 = 0 (1)

2

06 = 0 (1), 56 = 5 (3)

07 = 0 (1)

08 = 0 (1), 78 = 7 (3), 258 = 21 (1)

09 = 0 (1), 179 = 16 (2)

3

06 = 0 (1), 2536 = 105 (1)

07 = 0 (1), 2647 = 144 (2)

08 = 0 (1), 3748 = 252 (1)

09 = 0 (1), 3859 = 320 (2)

4

06 = 0 (1), 15546 = 430 (6)

07 = 0 (1), 30547 = 1068 (3)

08 = 0 (1), 17768 = 1022 (5), 30658 = 1589 (3)

09 = 0 (1), 30769 = 2256 (3), 50749 = 3712 (3)

5

06 = 0 (1), 315336 = 4305 (2), 415326 = 5600 (1)

07 = 0 (1), 406537 = 9936 (5)

08 = 0 (1), 427448 = 17892 (2), 517538 = 21483 (4)

09 = 0 (1), 528549 = 34960, (5), 628539 = 41520 (1)

6

06 = 0 (1), 2055446 = 16840 (3), 3255236 = 27195 (1), 4204326 = 33860 (1), 5304216 = 42925 (1)

07 = 0 (1), 3205447 = 55000 (6)

08 = 0 (1), 3106658 = 102837 (3), 4377348 = 147420 (1), 6406328 = 213402 (1)

09 = 0 (1), 3088769 = 183696 (14)

7

06 = 0 (1), 44054126 = 218960 (2)

07 = 0 (1), 53065327 = 640992 (6)

08 = 0 (1), 54076338 = 1445787 (7), 64076328 = 1707930 (4), 64175328 = 1711962 (1)

09 = 0 (1), 65186339 = 3496800 (2)

8

06 = 0 (1), 311044436 = 895275 (7), 421044506 = 1221870 (2), 431553226 = 1275170 (1), 531044216 = 1548445 (2), 553042016 = 1657225 (1)

07 = 0 (1), 421664437 = 3562968 (6)

08 = 0 (1), 311066658 = 6589877 (3), 641066328 = 13667738 (3), 753064218 = 16092433 (1)

09 = 0 (1), 652885339 = 31531872 (1), 654074339 = 31596672 (4)

9

06 = 0 (1), 3325552236 = 6018495 (1), 4410544126 = 7892360 (2)

07 = 0 (1), 5306665327 = 31412208 (11)

08 = 0 (1), 4437773348 = 76545756 (1), 5430764338 = 93093147 (5), 6440763328 = 110132442 (5), 6510766228 = 111443346 (4)

09 = 0 (1), 6418886439 = 278474880 (12), 7531865329 = 326952560 (1)

10

06 = 0 (1), 42110444326 = 44002820 (2), 43210443226 = 45962330 (1), 44215533126 = 47681900 (1), 53110444216 = 55760125 (2), 53310442216 = 56319925 (1), 55310442016 = 59679145 (2), 55530420016 = 60331825 (1)

07 = 0 (1), 53166664327 = 220709400 (2), 54310553227 = 227429400 (2), 54406652227 = 228238488 (2)

08 = 0 (1), 31110666658 = 421817781 (3), 64110666328 = 874802586 (3), 64310664328 = 878996762 (3), 75310664218 = 1029991697 (3), 77530642018 = 1068263553 (1)

09 = 0 (1), 52888888549 = 2066242576 (5), 64418864439 = 2516902752 (4), 66328855239 = 2598744000 (1)

 

Cifre \ base

10

11

12

13

1

0 (1)

011 = 0 (1)

012 = 0 (1)

013 = 0 (1)

2

0 (1), 9 (5)

011 = 0 (1), 3711 = 40 (1)

012 = 0 (1), B12 = 11 (6)

013 = 0 (1), 1B13 = 24 (3)

3

0 (1), 495 (1)

011 = 0 (1), 4A611 = 600 (2)

012 = 0 (1), 5B612 = 858 (1)

013 = 0 (1), 5C713 = 1008 (2)

4

0 (1), 6174 (1)

011 = 0 (1), 309811 = 4100 (5), 509611 = 6760 (5)

012 = 0 (1) , 3BB812 = 6908 (3), 419812 = 7172 (6)

013 = 0 (1), 30BA13 = 6744 (6), 50B813 = 11136 (3), 529813 = 11448 (6)

5

0 (1), 53955 (2), 61974 (4), 62964 (4)

011 = 0 (1), 72A7411 = 106440 (4)

012 = 0 (1), 64B6612 = 132990 (2), 83B7412 = 172744 (1)

013 = 0 (1), 83C8513 = 237216 (4)

6

0 (1), 420876 (7), 549945 (1), 631764 (1)

011 = 0 (1), 53187611 = 851560 (4)

012 = 0 (1), 420A9812 = 1038356 (10), 65BB5612 = 1617330 (1)

013 = 0 (1), 951A7413 = 3488424 (1)

7

0 (1), 7509843 (8)

011 = 0 (1), 631A87511 = 11141520 (14), 751A85411 = 13235160 (4)

012 = 0 (1), 841B97412 = 24924328 (9), 962B85312 = 28428543 (1)

013 = 0 (1), 951CA7413 = 45354456 (12), 972C95413 = 46125408 (2), A71CA5313 = 50923824 (2)

8

0 (1), 43208766 (7), 63317664 (1), 64308654 (3), 97508421 (1)

011 = 0 (1), 5321887611 = 103098980 (4), 7431876411 = 144005800 (1), 7521885411 = 145616420 (7)

012 = 0 (1), 4210AA9812 = 149566868 (24), 873BB74412 = 308551012 (1), A850A63212 = 383468294 (1)

013 = 0 (1), 641CCA8713 = 396540528 (4), 9541A87413 = 590407848 (1)

9

0 (1), 554999445 (1), 753098643 (14), 864197532 (1)

011 = 0 (1), 871AAA83311 = 1854824040 (7), 9751A853211 = 2074816680 (1)

012 = 0 (1), 665BBB55612 = 2812795986 (1), 8620BA95412 = 3661063120 (5), 8640BA75412 = 3667034800 (5), 9630BA85312 = 4094030655 (13)

013 = 0 (1), 9620CBA6413 = 7728089880 (5), A751CA75313 = 8621418120 (5), A861CA64313 = 8688993264 (5)

10

0 (1), 4332087666 (7), 6333176664 (1), 6431088654 (3), 6433086654 (3), 6543086544 (3), 9751088421 (3), 9753086421 (1), 9975084201 (1)

011 = 0 (1), 532218887611 = 12475622440 (4), 743218876411 = 17425363740 (2), 7631AA874411 = 17853788660 (5), 8621AA884311 = 20192249300 (11)

012 = 0 (1), 64310AA87612 = 32789316186 (4), 87650A654412 = 44518253956 (3), 8843BB773412 = 44873359960 (1), A8510AA63212 = 55220027654 (5)

013 = 0 (1), 86661A666513 = 90136502448 (9), 95441A887413 = 99790125192 (1), 99650B763413 = 103183026600 (5), A8630B964313 = 112962146040 (4)

 

Cifre \ base

14

15

16

17

1

014 = 0 (1)

015 = 0 (1)

016 = 0 (1)

017 = 0 (1)

2

014 = 0 (1), D14 = 13 (4), 2B14 = 39 (2), 4914 = 65 (1)

015 = 0 (1)

016 = 0 (1), F16 = 15 (4), 2D16 = 45 (4)

017 = 0 (1), 1F17 = 32 (3), 5B17 = 96 (1)

3

014 = 0 (1), 6D714 = 1365 (1)

015 = 0 (1), 6E815 = 1568 (2)

016 = 0 (1), 7F816 = 2040 (1)

017 = 0 (1), 7G917 = 2304 (2)

4

014 = 0 (1), 61B814 = 16822 (3)

015 = 0 (1), 30DC15 = 10332 (4), 50DA15 = 17080 (4), 52BA15 = 17500 (2), 70D815 = 23828 (4), 72B815 = 24248 (4), 92B615 = 30996 (1), B0D415 = 37324 (2)

016 = 0 (1), 1FFE16 = 8190 (6), 30ED16 = 12525 (4), 3FFC16 = 16380 (3), 52CB16 = 21195 (2)

017 = 0 (1), 30FE17 = 15008 (4), 50FC17 = 24832 (4), 52DC17 = 25376 (2), 70FA17 = 34656 (4), 72DA17 = 35200 (4), 90F817 = 44480 (4), 92D817 = 45024 (4), D0F417 = 64128 (2)

5

014 = 0 (1), 75D7714 = 285285 (2), 93D9514 = 356655 (4)

015 = 0 (1), A4E9515 = 523040 (1)

016 = 0 (1), 86F8816 = 552840 (2), A3FB616 = 671670 (3), A4FA616 = 675750 (4)

017 = 0 (1), B4GB617 = 943200

6

014 = 0 (1), 530CA914 = 2806869 (12), 76DD6714 = 4033575 (1), A40C9414 = 5534386 (2), A80C5414 = 5687994 (2)

015 = 0 (1), 41EECB15 = 3138716 (9), 960D8615 = 7141176 (3)

016 = 0 (1), 310EED16 = 3215085 (4), 532CCB16 = 5450955 (2), 840EB816 = 8654520 (5), 87FF7816 = 8912760 (1), A80E7616 = 11013750 (5), C60E9416 = 12979860 (4)

017 = 0 (1), D91E7417 = 19218912 (1)

7

014 = 0 (1), A62DA7414 = 78636870 (8)

015 = 0 (1), A62EB8515 = 118613600 (8), B70ED7415 = 130662784 (10)

016 = 0 (1), B62FC9516 = 191036565 (9), B71FD8516 = 192019845 (11), C83FB7416 = 209976180 (1)

017 = 0 (1), B72GD9617 = 275701824 (4), C93GC7517 = 302762304 (2), D92GD7417 = 326816640 (2)

8

014 = 0 (1), 7550C88714 = 778265943 (8), A410CC9414 = 1084826418 (6), A810CC5414 = 1114944506 (6), CA70C63214 = 1344056324 (1)

015 = 0 (1), 861EEC8715 = 1436736952 (22), 9660D88615 = 1610680176 (3), A940DA5515 = 1814193080 (4), A94EE95515 = 1814904980 (1)

016 = 0 (1), 3110EEED16 = 823193325 (4), 5332CCCB16 = 1395838155 (2), 7530ECA916 = 1966140585 (4), A832CC7616 = 2821901430 (8), C610EE9416 = 3322998420 (4), C630EC9416 = 3325095060 (4), C650EA9416 = 3327191700 (4), CA10EE5416 = 3390107220 (4)

017 = 0 (1), 951GGEB817 = 3816574944 (4), A940FC7717 = 4326381568 (5), D91GGE7417 = 5554479840 (3)

9

014 = 0 (1), 776DDD66714 = 11121123195 (1), A731DBA6414 = 15519443160 (8), B852DA85314 = 17116239465 (1)

015 = 0 (1), A742EBA7515 = 26872751360 (8), B862EB86415 = 29629282144 (6), C962EB85315 = 32363032128 (1)

016 = 0 (1), 887FFF77816 = 36641437560 (1), C851FDA7416 = 53773064820 (7), C862FC97416 = 53790886260 (6), CA40FEB5416 = 54292114260 (8), DA72FC85316 = 58639501395 (1)

017 = 0 (1), C962GDA7517 = 87551205696 (6), C963GCA7517 = 87552620640 (7), EB82GD85317 = 102371672448 (1)

10

014 = 0 (1), A4110CCC9414 = 212627063506 (6), A4310CCA9414 = 212837890122 (6), A7410CC96414 = 217370670556 (4), A8110CCC5414 = 218530219674 (6), A8310CCA5414 = 218741046290 (6), CA710CC63214 = 263436371172 (6)

015 = 0 (1), 96660D888615 = 362461765176 (3), AA54EE994515 = 410973731840 (1), B761EEC87415 = 441865116784 (6)

016 = 0 (1), 31110EEEED16 = 210739588845 (4), 53332CCCCB16 = 357340859595 (2), 75310EECA916 = 503334235305 (4), 75330ECCA916 = 503367781545 (4), 75530ECAA916 = 503904651945 (4), 86530ECA9816 = 576919095960 (5), A8532CCA7616 = 722949950070 (8), A8732CC87616 = 723486820470 (8), C6110EEE9416 = 850689715860 (4), C6310EEC9416 = 851226586260 (4), C6330ECC9416 = 851260132500 (4), C6510EEA9416 = 851763456660 (4), C6530ECA9416 = 851797002900 (4), C6550EAA9416 = 851830549140 (4), CA110EEE5416 = 867869584980 (4), CA310EEC5416 = 868406455380 (4), CA510EEA5416 = 868943325780 (4), CB610EE94416 = 873506728260 (4), FDB70E842116 = 1089697907745 (1)

017 = 0 (1), 6441GGECCB17 = 741119907520 (24)

 

Cifre \ base

18

19

20

1

018 = 0 (1)

019 = 0 (1)

020 = 0 (1)

2

018 = 0 (1), H18 = 17 (9)

019 = 0 (1), 3F19 = 72 (2)

020 = 0 (1), J20 = 19 (6), 2H20 = 57 (3), 6D20 = 133 (1)

3

018 = 0 (1), 8H918 = 2907 (1)

019 = 0 (1), 8IA19 = 3240 (2)

020 = 0 (1), 9JA20 = 3990 (1)

4

018 = 0 (1), 61FC18 = 35598 (3) , A1F818 = 58922 (3)

019 = 0 (1), 30HG19 = 20916 (9), 50HE19 = 34632 (9), 52FE19 = 35316 (9) 74DC19 = 49716 (9)

020 = 0 (1), 52GF20 = 41135 (6), C3F820 = 97508 (1)

5

018 = 0 (1), 97H9918 = 991287 (2), C5HB618 = 1294584 (1)

019 = 0 (1), C4ID719 = 1598040 (3), C5IC719 = 1604880 (4)

020 = 0 (1), A8JAA20 = 1671810 (2), D5JD720 = 2127867 (4)

6

018 = 0 (1), 820GFA18 = 15331960 (12), 851FCA18 = 15652342 (7), 98HH8918 = 17950725 (1), A60GB818 = 19530926 (4), A62EB818 = 19541942 (3)

019 = 0 (1), 531GFE19 = 12784392 (6), 72IIFC19 = 17723592 (4), 863ECB19 = 20616588 (4), B51GD819 = 27901584 (5), F40HE419 = 37669176 (2)

020 = 0 (1), 840IFC20 = 26247512 (17), 842GFC20 = 26262712 (5), A42GFA20 = 32662710 (6), A9JJ9A20 = 33599790 (1), E82GB620 = 46102626 (1), EC0I7620 = 46727346 (3)

7

018 = 0 (1), D82HE9518 = 457589255 (4)

019 = 0 (1), E83IEA519 = 678970800 (14)

020 = 0 (1), E84JEB620 = 922397826 (13), FA4JE9520 = 992797785 (1)

8

018 = 0 (1), 8320GFEA18 = 5003674498 (15), 8632EEBA18 = 5107798648 (32), 8652ECBA18 = 5111577136 (7), A610GGB818 = 6328261934 (4), A760GBA818 = 6371720360 (4), CB5HHB6618 = 7732110102 (1)

019 = 0 (1), 8663ECCB19 = 7448597352 (4) B651GDC819 = 10127489904 (5), CA41GE8719 = 11207069352 (5)

020 = 0 (1), 8540IFEC20 = 10572950292 (11), G650IED420 = 20880149864 (2)

9

018 = 0 (1), 998HHH88918 = 104995732401 (1), C842HED9618 = 137278985820 (14), EA42HED7418 = 160543346998 (2), EA62HEB7418 = 160611370798 (12)

019 = 0 (1), FB52IFD7419 = 264828669840 (2), FB73IEB7419 = 264925230120 (1)

020 = 0 (1), AA9JJJ99A20 = 269439995790 (1), EA62JGD9620 = 371593573386 (7), FA73JFC9520 = 397260764985 (7), GC62JGD7420 = 425353573344 (2), GC83JFB7420 = 425484764544 (1)

10

018 = 0 (1), A6110GGGB818 = 2050360677710 (4), A7320GFEA818 = 2062639084076 (8), A7610GGBA818 = 2064441736808 (4), A7760GBAA818 = 2065223988476 (4), A8750GCA9818 = 2076209942642 (4), C7652ECBA618 = 2461299912294 (1), CC65HHBB5618 = 2516428367700 (1), E5430GEDC418 = 2834682550528 (2), HFD90G842118 = 3545674042933 (1)

019 = 0 (1), 86663ECCCB19 = 2689057805868 (4), B6651GDCC819 = 3657069168624 (5), B761IIGCB819 = 3673906923024 (10), CA641GEC8719 = 4047644079468 (5), FBA41GE87419 = 5036266221336 (1)

020 = 0 (1), 86510IIEDC20 = 4256067029872 (23), A5410IIFEA20 = 5253187030290 (15), A9542GFEAA20 = 5357065085810 (16), C6532GGED820 = 6304201093868 (16)