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, 321065449877 is a pandigital number in base 10. The first few pandigital base 10 numbers are given by (sequence A050278 in OEIS): The On-Line Encyclopedia of Integer Sequences (OEIS) is a web-based searchable database of integer sequences. ...

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

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

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

A049363 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 with the exception of 0 and numbers of the form 2ⁿ − 1 (the Mersenne numbers). The larger the base, the rarer pandigital numbers become, though one can always find runs of 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. In mathematics, a Mersenne prime is a prime number that is one less than a power of two. ...

Conversely, the smaller the base, the fewer pandigital numbers without redundant digits there are. 2 is the only such pandigital number in base 2, while there are more of these in base 10.

Sometimes, the term is used to refer only to pandigital numbers with no redundant digits. And in some cases, a number might be called pandigital even if it doesn't have a zero as a significant digit, for example, 923456781 (these are sometimes referred to as "zeroless pandigital numbers").

No base 10 pandigital number can be a prime number if it doesn't have redundant digits. The sum of the digits 0 to 9 is 45, passing the test for divisibility for both 3 and 9. The first base 10 pandigital prime is 10123457689, A050288 lists more. In mathematics, a prime number (or prime) is a natural number greater than one whose only positive divisors are one and itself. ...

For different reasons, redundant digits are also required for a pandigital number (in any base except unary) to also be a palindromic number in that base. The smallest pandigital palindromic number in base 10 is 12345678987654321. A palindromic number is a symmetrical number like 16461, that remains the same when its digits are reversed. ...

The largest pandigital number without redundant digits to be also a square number is 9814072356. In mathematics, a square number, sometimes also called a perfect square, is an integer that can be written as the square of some other integer. ...

Currently, two zeroless pandigital Friedman numbers are known: 123456789 = ((86 + 2 * 7)⁵ - 91) / 34, and 987654321 = (8 * (97 + 6/2)⁵ + 1) / 3⁴. A Friedman number is an integer which in a given base is the result of an expression using its own digits in combination with any of the four basic arithmetic operators (+, -, ×, ÷) and sometimes exponentiation. ...

While much of what has been said does not apply to Roman numerals, there are pandigital numbers: MCDXLIV, MCDXLVI, MCDLXIV, MCDLXVI, MDCXLIV, MDCXLVI, MDCLXIV, MDCLXVI A105416 use each of the digits just once, while A105417 has pandigital Roman numerals with repeats. The system of Roman numerals is a numeral system originating in ancient Rome, and was adapted from Etruscan numerals. ...