Integers with restricted digits are nonnegative integers whose expansions in a fixed integer base 
use only digits from a prescribed nonempty set 
.
The usual expansion without leading zeros is intended. The expansion of 0 is the
single digit 0, so it qualifies exactly when 
. Such restrictions are studied in analytic
number theory (Banks and Shparlinski 2004).
For a positive integer 
, the number of qualifying positive
integers with exactly 
digits is
 |
The first digit can be any nonzero member of 
, and each remaining digit can be
any member.
Write 
for the set of nonnegative
integers using only the digits 0 and 1 in base 
.
This condition is automatic in base 2 but restrictive in
larger bases. For example, 
(Numberphile 2015). It is conjectured that 
(OEIS A146025;
Burrell and Yu 2021). Completeness of this list has not been proved.
Burrell and Yu (2021) proved that, for every 
, there is a constant 
such that, for every positive integer 
,
 |
Known members of 
include 0, 1, 5, 16400, 16401, 16405, 82000,
82001, and 82005 (OEIS A263684). The same bound holds after imposing the additional base
3 restriction. The estimate does not prove that either intersection is finite or
that the conjectured list is complete.
