if it is pandigital
and all its multiples
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
Pictorial representation of remainders (mod 2, 3, ...,11) frequency. For a table of values and more details click here
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