A pandigital number is a nonnegative integer whose decimal expansion contains each of the digits from 0 to 9, with no leading zeros. A zeroless pandigital number instead contains all the digits 1 through 9 and excludes 0. A restricted pandigital number has each of the required digits exactly once. For example, 6729/13458 is a (zeroless, restricted) pandigital fraction and 1023456789 is the smallest (zerofull) pandigital number.
A number of interesting pandigital pi approximations are known.
The first few zerofull restricted pandigital numbers are 1023456789, 1023456798, 1023456879, 1023456897, 1023456978, ... (OEIS A050278). A 10-digit pandigital number is always divisible by 9 since
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(1)
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This passes the divisibility test for 9 since 
.
The smallest unrestricted pandigital primes must therefore have 11 digits (no two of which can be 0). The first few unrestricted pandigital primes are therefore 10123457689, 10123465789, 10123465897, 10123485679, ... (OEIS A050288).
If zeros are excluded, the first few "zeroless" restricted pandigital numbers are 123456789, 123456798, 123456879, 123456897, 123456978, 123456987, ... (OEIS A050289), and the first few zeroless pandigital primes are 1123465789, 1123465879, 1123468597, 1123469587, 1123478659, ... (OEIS A050290).
The sum of the first 32423 (a palindromic number) consecutive primes is 5897230146, which is restricted pandigital (G. L. Honaker, Jr., pers. comm.). No other palindromic number shares this property.
The positive integers 
for which the Fibonacci recurrence
relation
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(2)
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with 
and 
has a restricted zeroless pandigital term are 718, 1790, 1993, 2061, 2259, 3888,
3960, 4004, 4396, 5093, 5832, 7031, 7310, 7712, 8039, 8955, 9236, ....
The products of these restricted pandigital numbers satisfy
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(3)
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(A. Povolotsky, pers. comm., Aug. 27, 2022).
