There are some extremely large Tuesday numbers, with billions of digits. Probably there is no bound on how large they can get, though nobody knows how to prove this.
If $n$ is a positive integer, the decimal expansion of $1/n$ is eventually periodic, and the period has at most $n-1$ digits. The period can be exactly $n-1$ digits only if $n$ is a prime number (though this does not happen for every prime). For example,
has period $7-1=6$ (and indeed $7$ is prime).
If the decimal expansion of $1/p$ has period $p-1$, we call $p$ a full reptend prime.
We can obtain a Tuesday number (also called a cyclic number) from one full period of the decimal expansion of the reciprocal of a full reptend prime, and every Tuesday number arises this way. For example, $7$ is a full reptend prime, so $142857$ is a Tuesday number:
3\cdot 142857=42857\>1,&4\cdot 142857=57\>1428,\\
5\cdot 142857=7\>14285,& 6\cdot 142857=857\>142.\\
It can be proven that $p$ is a reptend prime if and only if the first $p-1$ powers of $10$ give distinct remainders when divided by $p$ (in this case one says $10$ is a primitive root modulo $p$). Artin's conjecture on primitive roots asserts that every non-square positive integer is a primitive root modulo $p$ for infinitely many primes $p$, and this would imply that there exist arbitrarily large Tuesday numbers.