A pandigital number n whose multiples up to k ⋅ n, with k >1, are pandigital as well. more
The first 600 persistent numbers :
1023456789,
1023456879,
1023457689,
1023457869,
1023458679,
1023458769,
1023465789,
1023465879,
1023467589,
1023467859,
1023468579,
1023468759,
1023475689,
1023475869,
1023476589,
1023476859,
1023478569,
1023478659,
1023485679,
1023485769,
1023486579,
1023486759,
1023487569,
1023487659,
1023567849,
1023567948,
1023567984,
1023568479,
1023569478,
1023569784,
1023576849,
1023576948,
1023576984,
1023578469,
1023579468,
1023579684,
1023584679,
1023584769,
1023594678,
1023594768,
1023596784,
1023597684,
1023657849,
1023657948,
1023657984,
1023658479,
1023659478,
1023659784,
1023675849,
1023675948,
1023675984,
1023678459,
1023679458,
1023679584,
1023684579,
1023684759,
1023694578,
1023694758,
1023695784,
1023697584,
1023756849,
1023756948,
1023756984,
1023758469,
1023759468,
1023759684,
1023765849,
1023765948,
1023765984,
1023768459,
1023769458,
1023769584,
1023784569,
1023784659,
1023794568,
1023794658,
1023795684,
1023796584,
1023845679,
1023845769,
1023846579,
1023846759,
1023847569,
1023847659,
1023945678,
1023945768,
1023946578,
1023946758,
1023947568,
1023947658,
1023956784,
1023957684,
1023965784,
1023967584,
1023975684,
1023976584,
1024356798,
1024356978,
1024357698,
1024357968,
1024359678,
1024359768,
1024365798,
1024365978,
1024367598,
1024367958,
1024369578,
1024369758,
1024375698,
1024375968,
1024376598,
1024376958,
1024379568,
1024379658,
1024395678,
1024395768,
1024396578,
1024396758,
1024397568,
1024397658,
1024567839,
1024567893,
1024567938,
1024568379,
1024568793,
1024569378,
1024576839,
1024576893,
1024576938,
1024578369,
1024578693,
1024579368,
1024583679,
1024583769,
1024586793,
1024587693,
1024593678,
1024593768,
1024657839,
1024657893,
1024657938,
1024658379,
1024658793,
1024659378,
1024675839,
1024675893,
1024675938,
1024678359,
1024678593,
1024679358,
1024683579,
1024683759,
1024685793,
1024687593,
1024693578,
1024693758,
1024756839,
1024756893,
1024756938,
1024758369,
1024758693,
1024759368,
1024765839,
1024765893,
1024765938,
1024768359,
1024768593,
1024769358,
1024783569,
1024783659,
1024785693,
1024786593,
1024793568,
1024793658,
1024835679,
1024835769,
1024836579,
1024836759,
1024837569,
1024837659,
1024856793,
1024857693,
1024865793,
1024867593,
1024875693,
1024876593,
1024935678,
1024935768,
1024936578,
1024936758,
1024937568,
1024937658,
1025673489,
1025673849,
1025673948,
1025673984,
1025674398,
1025674839,
1025674893,
1025674938,
1025683749,
1025683947,
1025683974,
1025684397,
1025684739,
1025684937,
1025687349,
1025687493,
1025689347,
1025689473,
1025689734,
1025689743,
1025693487,
1025693748,
1025693847,
1025694738,
1025694837,
1025694873,
1025697384,
1025697438,
1025698374,
1025698437,
1025698734,
1025698743,
1025734689,
1025734869,
1025736849,
1025736948,
1025736984,
1025738469,
1025739468,
1025739684,
1025743698,
1025743968,
1025746839,
1025746893,
1025746938,
1025748369,
1025748693,
1025749368,
1025836749,
1025836947,
1025836974,
1025837469,
1025839467,
1025839674,
1025843697,
1025843967,
1025846739,
1025846937,
1025847369,
1025849367,
1025867349,
1025867493,
1025869347,
1025869473,
1025869734,
1025869743,
1025873469,
1025874693,
1025893467,
1025894673,
1025896734,
1025896743,
1025934687,
1025934867,
1025936748,
1025936847,
1025937468,
1025938467,
1025946738,
1025946837,
1025946873,
1025947368,
1025948367,
1025948673,
1025967384,
1025967438,
1025968374,
1025968437,
1025968734,
1025968743,
1025973684,
1025974368,
1025983674,
1025984367,
1025986734,
1025986743,
1026573489,
1026573849,
1026573948,
1026573984,
1026574398,
1026574839,
1026574893,
1026574938,
1026583749,
1026583947,
1026583974,
1026584397,
1026584739,
1026584937,
1026587349,
1026587493,
1026589347,
1026589473,
1026589734,
1026589743,
1026593487,
1026593748,
1026593847,
1026594738,
1026594837,
1026594873,
1026597384,
1026597438,
1026598374,
1026598437,
1026598734,
1026598743,
1026734589,
1026734859,
1026735849,
1026735948,
1026735984,
1026738459,
1026739458,
1026739584,
1026743598,
1026743958,
1026745839,
1026745893,
1026745938,
1026748359,
1026748593,
1026749358,
1026835749,
1026835947,
1026835974,
1026837459,
1026839457,
1026839574,
1026843597,
1026843957,
1026845739,
1026845937,
1026847359,
1026849357,
1026857349,
1026857493,
1026859347,
1026859473,
1026859734,
1026859743,
1026873459,
1026874593,
1026893457,
1026894573,
1026895734,
1026895743,
1026934587,
1026934857,
1026935748,
1026935847,
1026937458,
1026938457,
1026945738,
1026945837,
1026945873,
1026947358,
1026948357,
1026948573,
1026957384,
1026957438,
1026958374,
1026958437,
1026958734,
1026958743,
1026973584,
1026974358,
1026983574,
1026984357,
1026985734,
1026985743,
1027345689,
1027345869,
1027346589,
1027346859,
1027348569,
1027348659,
1027356849,
1027356948,
1027356984,
1027358469,
1027359468,
1027359684,
1027365849,
1027365948,
1027365984,
1027368459,
1027369458,
1027369584,
1027384569,
1027384659,
1027394568,
1027394658,
1027395684,
1027396584,
1027435698,
1027435968,
1027436598,
1027436958,
1027439568,
1027439658,
1027456839,
1027456893,
1027456938,
1027458369,
1027458693,
1027459368,
1027465839,
1027465893,
1027465938,
1027468359,
1027468593,
1027469358,
1027483569,
1027483659,
1027485693,
1027486593,
1027493568,
1027493658,
1028356749,
1028356947,
1028356974,
1028357469,
1028359467,
1028359674,
1028365749,
1028365947,
1028365974,
1028367459,
1028369457,
1028369574,
1028374569,
1028374659,
1028394567,
1028394657,
1028395674,
1028396574,
1028435697,
1028435967,
1028436597,
1028436957,
1028439567,
1028439657,
1028456739,
1028456937,
1028457369,
1028459367,
1028465739,
1028465937,
1028467359,
1028469357,
1028473569,
1028473659,
1028493567,
1028493657,
1028567349,
1028567493,
1028569347,
1028569473,
1028569734,
1028569743,
1028573469,
1028574693,
1028593467,
1028594673,
1028596734,
1028596743,
1028657349,
1028657493,
1028659347,
1028659473,
1028659734,
1028659743,
1028673459,
1028674593,
1028693457,
1028694573,
1028695734,
1028695743,
1028734569,
1028734659,
1028745693,
1028746593,
1028934567,
1028934657,
1028945673,
1028946573,
1028956734,
1028956743,
1028965734,
1028965743,
1029345687,
1029345867,
1029346587,
1029346857,
1029348567,
1029348657,
1029356748,
1029356847,
1029357468,
1029358467,
1029365748,
1029365847,
1029367458,
1029368457,
1029374568,
1029374658,
1029384567,
1029384657,
1029456738,
1029456837,
1029456873,
1029457368,
1029458367,
1029458673,
1029465738,
1029465837,
1029465873,
1029467358,
1029468357,
1029468573,
1029473568,
1029473658,
1029483567,
1029483657,
1029485673,
1029486573,
1029567384,
1029567438,
1029568374,
1029568437,
1029568734,
1029568743,
1029573684,
1029574368,
1029583674,
1029584367,
1029586734,
1029586743,
1029657384,
1029657438,
1029658374,
1029658437,
1029658734,
1029658743,
1029673584,
1029674358,
1029683574,
1029684357,
1029685734,
1029685743,
1029735684,
1029736584,
1029743568,
1029743658,
1029835674,
1029836574,
1029843567,
1029843657,
1029856734,
1029856743,
1029865734,
1029865743,
1032456789,
1032456879,
1032457689,
1032457869,
1032458679,
1032458769,
1032465789,
1032465879,
1032467589,
1032467859,
1032468579,
1032468759,
1032475689,
1032475869,
1032476589,
1032476859,
1032478569,
1032478659,
1032485679,
1032485769,
1032486579,
1032486759,
1032487569,
1032487659.
Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 19422720 values, from 1023456789 to 99876543210).
n\r | 0 | 1 |
2 | 9792576 | 9630144 | 2 |
3 | 7925760 | 5592960 | 5904000 | 3 |
4 | 4817376 | 4935456 | 4975200 | 4694688 | 4 |
5 | 4675968 | 3686688 | 3686688 | 3686688 | 3686688 | 5 |
6 | 3993120 | 2854944 | 3061440 | 3932640 | 2738016 | 2842560 | 6 |
7 | 2774416 | 2774028 | 2774095 | 2774661 | 2775132 | 2775605 | 2774783 | 7 |
8 | 2394324 | 2460048 | 2524572 | 2391912 | 2423052 | 2475408 | 2450628 | 2302776 | 8 |
9 | 3911040 | 1578240 | 1578240 | 1578240 | 1578240 | 1889280 | 2436480 | 2436480 | 2436480 | 9 |
10 | 2419200 | 1695744 | 1695744 | 1695744 | 1695744 | 2256768 | 1990944 | 1990944 | 1990944 | 1990944 | 10 |
11 | 1796700 | 1800724 | 1743000 | 1737248 | 1758876 | 1750272 | 1734808 | 1748536 | 1778112 | 1782952 | 1791492 |
A pictorial representation of the table above
Imagine to divide the members of this family by a number n and compute the remainders. Should they be uniformly distributed, each remainder from 0 to n-1 would be obtained in about (1/n)-th of the cases. This outcome is represented by a white square. Reddish (resp. bluish) squares represent remainders which appear more (resp. less) frequently than 1/n.