In base 10, the number 3816547290 contains every digit exactly once. When I take the first N digits, that substring is divisible by N. For example, 381 is divisible by 3, 38165 is divisible by 5, etc. In base 10, 3816547290 is the only such number which meets the above criteria.
My question is, how can I find these numbers for any given base? I made a program to search for them up to base 30. Out of all the bases it tried, it only found numbers for bases 2, 4, 6, 8, 10, and 14. These numbers where as follows (I use digits 0-9 and A-D):
base 2 - 10 base 4 - 1230 base 4 - 3210 base 6 - 143250 base 6 - 543210 base 8 - 32541670 base 8 - 52347610 base 8 - 56743210 base 10 - 3816547290 base 14 - 9C3A5476B812D0
I do not know if any such numbers exist in bases larger than 14. Even if such numbers do exist, I don't know how to find them. Can anybody think of a general algorithm for finding such numbers or a general proof that they do not exist?
UPDATE: my code to find said numbers can be found on Github if anybody wants to look at it. Essentially, the program uses recursion to try all permutations of the digits. If it explores a set of digits which themselves do not meet the criterion, it can stop searching for permutations with that prefix (i.e. if it got to 382 in base 10, it would realize that that number is not divisible by 3 and would thus not search anymore numbers which start with 382).