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persistent numbers
A number    is called  -persistent if it is pandigital and all its multiples    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  -persistent numbers for each  , but there is not a  -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  -persistent numbers, with  , are 1023456789, 1023456879, 1023457689, 1023457869, 1023458679, 1023458769, 1023465789, 1023465879, 1023467589, 1023467859 more terms

K-persistent numbers, with k≥2 can also be... (you may click on names or numbers and on + to get more values)