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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 + 90216475538 91352476800 96024519378 Achilles 1324890675 + 97503981624 97940123568 98175243600 amicable 12970438065 84590271368 binomial 1520843976 + 96087416253 96728953041 97486512903 c.decagonal 15208439761 + 84922376501 86534247901 92014786531 c.heptagonal 10892467538 + 79842085163 83609154728 93207845186 c.nonagonal 11930258746 + 51962014378 68013542971 76895321041 c.octagonal 1532487609 + 91625473809 94618375201 96438197025 c.pentagonal 11835492076 + 85670296431 92380451676 98426737051 c.square 12530869741 + 92801643745 93486177205 97738142065 c.triangular 10751862349 + 87134152960 91274683510 98017361254 cake 37054129568 97694358210 canyon 1023456789 + 98765432107 98765432108 98765432109 Carmichael 17392546081 61280451937 Cunningham 1608972543 + 96843572810 97781036542 99241380675 decagonal 1089346527 + 96805143792 98604321175 98941230675 deceptive 10214758369 + 50456182739 51829603741 61280451937 eRAP 7269813504 + 46270415938 52104684379 58028163749 esthetic 9876543210 10123456789 89876543210 98765432101 gapful 1023458769 + 99876542310 99876543120 99876543210 Harshad 1067283945 + 9876542310 9876543120 9876543210 heptagonal 1423809765 + 81539726104 87925034617 98630421751 hex 11342506897 + 83976402517 89456701327 97368012541 hexagonal 1520843976 + 91280467356 96087416253 97486512903 Hogben 10158926473 + 78567249103 79146850231 79865303421 katadrome 9876543210 Lehmer 14062987315 + 95138267401 98237406751 98367501241 magic 51646873290 modest 1026974358 + 1845062937 1854926073 1872506493 mountain 1234567890 + 78986543210 79876543210 89876543210 nialpdrome 9876543210 + 98776543210 98876543210 99876543210 nonagonal 3906714825 + 93980674125 96764235081 98243361750 octagonal 14993207685 + 81376952405 87491960325 91528477360 pancake 10457689132 + 93474286501 96728513204 97437506182 pandigital 1023456789 + 9876543120 9876543201 9876543210 pentagonal 5329461870 + 97893421605 98541327067 98678243510 Poulet 15076432489 + 51829603741 61280451937 92345876701 power 1532487609 + 94618375201 95371027684 96438197025 powerful 1324890675 + 97503981624 97940123568 98175243600 prime 10123457689 + 98876524301 98876530421 98876532401 primeval 10123456789 pronic 2965837140 + 84952137690 96317432850 97384252160 Proth 4296015873 + 92609183745 98615427073 98761703425 repunit 10158926473 + 78567249103 79146850231 79865303421 Ruth-Aaron 3518072649 + 98673140527 98673140528 99243756810 Saint-Exupery 3725948160 25719663840 61583799420 square 1532487609 + 94618375201 95371027684 96438197025 star 11456829037 + 86549104273 90261814537 94618251037 straight-line 9876543210 taxicab 4315087296 5480172936 20763541981 30025986417 tetrahedral 18259738640 26507314869 triangular 1520843976 + 96087416253 96728953041 97486512903 vampire 1023657984 + 7986341520 8036197425 8143697520 weakly prime 12695840537 + 89567803421 90684725371 95440286317