Is there any other way than just converting numbers between bases, to find out if a given number of one base represented in another arbitrary base is going to have certain count of unique digits?
I know this sounds cryptic, so let's take an example of base10 integer "365580" and assume we use the following set of numerals = "0123456789abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWXYZ" that would cover bases up to 62.
Decimal "365580" comprises of 6 digits, 5 of which are unique. When converted to let's say base13, it is "ca527" which is 5 digits and all 5 are unique. But when converted to base26 it is "kkkk" which is 4 digits with just 1 unique digit.
I am wondering if there is a way to tell by "looking" at a decimal "365580" that in base26 (or any other) it's going to be represented by a specific counts of unique digits. By "looking" I mean some operation that is computationally less expensive than converting between bases.