Pandigital number

In mathematics, a pandigital number is an integer that in a given base has among its significant digits each digit used in the base at least once. For example, 1223334444555567890 is a pandigital number in base 10. The first few pandigital base 10 numbers are given by (sequence in the OEIS):

1023456789, 1023456798, 1023456879, 1023456897, 1023456978, 1023456987, 1023457689

The smallest pandigital number in a given base b is an integer of the form

$b^{b-1}+\sum _{d=2}^{b-1}db^{b-1-d}$ $b^{b - 1} + \sum_{d = 2}^{b - 1} db^{b - 1 - d}$

The following table lists the smallest pandigital numbers of a few selected bases:

gives the base 10 values for the first 18 bases.

In a trivial sense, all positive integers are pandigital in unary (or tallying). In binary, all integers are pandigital except for 0 and numbers of the form $2^{n}-1$ $2^{n}-1$ (the Mersenne numbers). The larger the base, the rarer pandigital numbers become, though one can always find runs of $b^{x}$ $b^x$ consecutive pandigital numbers with redundant digits by writing all the digits of the base together (but not putting the zero first as the most significant digit) and adding x + 1 zeroes at the end as least significant digits.

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