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persistent numbers
A number  $n$  is called  $k$-persistent if it is pandigital and all its multiples  $2\cdot n, 3\cdot n,\dots, k\cdot n$  are pandigital as well.

In this context, a number is pandigital if it contains all the 10 digits at least once.

For example, the pandigital number 1023457869 is 2-persistent, because 2⋅1023457869 = 2046915738 is pandigital as well, but not 3-persistent, because 3⋅1023457869 = 3070373607.

As R. Honsberger proves in his book More Mathematical Morsels, there exist infinite  $k$-persistent numbers for each  $k$, but there is not a  $\infty$-persistent number.

Among the 10-digit number the highest persistency is 4, attained for example by 1053274689. Among the 11-digit number we reach 6 (48602175913) and among 12-digits numbers 8 (702483793156).

The first  $k$-persistent numbers, with  $k\ge2$, are 1023456789, 1023456879, 1023457689, 1023457869, 1023458679, 1023458769, 1023465789, 1023465879, 1023467589, 1023467859 more terms

Pictorial representation of remainders (mod 2, 3, ...,11) frequency. For a table of values and more details click here
remainders of k-persistent numbers, with k≥2

A graph displaying how many k-persistent numbers, with k≥2 are multiples of the primes p from 2 to 71. In black the ideal line 1/p.
divisibility of k-persistent numbers, with k≥2
Useful links Mathworld, Persistent Number
OEIS, Sequence A204047
K-persistent numbers, with k≥2 can also be... (you may click on names or numbers and on + to get more values)

ABA 3047618592 4163098752 7025391648 7654803912 + 96024519378 Achilles 1324890675 1854207963 3042571896 3047618592 + 98175243600 amicable 12970438065 84590271368 binomial 1520843976 2718609453 3574816290 3704581926 + 97486512903 c.decagonal 15208439761 24708692531 24869230751 27186094531 + 92014786531 c.heptagonal 10892467538 16450937728 19286104573 23401878956 + 93207845186 c.nonagonal 11930258746 12874106953 16209451378 16320914785 + 76895321041 c.octagonal 1532487609 3285697041 5102673489 8967143025 + 96438197025 c.pentagonal 11835492076 12519743806 15478913206 18035947266 + 98426737051 c.square 12530869741 13285967041 13978421605 14340896725 + 97738142065 c.triangular 10751862349 12467543089 13018597624 13970256814 + 98017361254 cake 37054129568 97694358210 Carmichael 17392546081 61280451937 Cunningham 1608972543 1635798024 3584297160 8423751960 + 99241380675 decagonal 1089346527 1632907485 1945802376 3946258170 + 98941230675 deceptive 10214758369 10365844729 12285306749 17392546081 + 61280451937 eRAP 7269813504 10698534927 18362709504 19563792480 + 58028163749 esthetic 9876543210 10123456789 89876543210 98765432101 gapful 1023458769 1023475869 1023567948 1023569478 + 99876543210 Harshad 1067283945 1067284395 1067284935 1067289345 + 9876543210 heptagonal 1423809765 1640538297 10839469752 13965804297 + 98630421751 hex 11342506897 13946150827 14528930617 14835762019 + 97368012541 hexagonal 1520843976 2718609453 3574816290 3842970615 + 97486512903 Hogben 10158926473 10865247933 12859673401 14238097653 + 79865303421 katadrome 9876543210 Lehmer 14062987315 14350972681 14352078469 14857693201 + 98367501241 magic 51646873290 modest 1026974358 1053287469 1083697524 1368025974 + 1872506493 nialpdrome 9876543210 98765432100 98765432110 98765432210 + 99876543210 nonagonal 3906714825 6754039821 12053194786 15299473806 + 98243361750 octagonal 14993207685 15539762408 17063925845 24379805416 + 91528477360 pancake 10457689132 11956378204 12268593047 12458390176 + 97437506182 pandigital 1023456789 1023456879 1023457689 1023457869 + 9876543210 pentagonal 5329461870 10981754362 12087934465 14678590432 + 98678243510 Poulet 15076432489 17392546081 25089467413 29384567041 + 92345876701 power 1532487609 2081549376 3285697041 3719048256 + 96438197025 powerful 1324890675 1532487609 1854207963 2081549376 + 98175243600 prime 10123457689 10123465789 10123485679 10123485769 + 98876532401 primeval 10123456789 pronic 2965837140 7932485160 8264537190 10476853092 + 97384252160 Proth 4296015873 6803947521 10277486593 12236750849 + 98761703425 repunit 10158926473 10865247933 12859673401 14238097653 + 79865303421 Ruth-Aaron 3518072649 5217860493 5380412697 8046513729 + 99243756810 Saint-Exupery 3725948160 25719663840 61583799420 square 1532487609 2081549376 3285697041 3719048256 + 96438197025 star 11456829037 12054876913 14082961537 14126598037 + 94618251037 straight-line 9876543210 taxicab 4315087296 5480172936 20763541981 30025986417 tetrahedral 18259738640 26507314869 triangular 1520843976 2718609453 3574816290 3704581926 + 97486512903 vampire 1023657984 1023679584 1043568297 1054273968 + 8143697520 weakly prime 12695840537 13508267249 21635702849 29473078651 + 95440286317