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persistent numbers
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 
297925769630144 2 
3792576055929605904000 3 
44817376493545649752004694688 4 
546759683686688368668836866883686688 5 
6399312028549443061440393264027380162842560 6 
72774416277402827740952774661277513227756052774783 7 
823943242460048252457223919122423052247540824506282302776 8 
9391104015782401578240157824015782401889280243648024364802436480 9 
102419200169574416957441695744169574422567681990944199094419909441990944 10 
1117967001800724174300017372481758876175027217348081748536177811217829521791492

A pictorial representation of the table above
motab
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.