I don't know about lookup speeds, but I believe the storage
requirement would be smaller if you go with a numeric data type over
characters:
To represent all 13-digit unsigned integers, you need 47 bits (2^47 =
1.4 * 10^14).
To represent a 13-character string (that may be composed of digits),
you need at least 104 bits (8 bytes * 13). You might need more than
one byte per character depending on your character encoding.
So it looks to me like storing numbers in numeric format requires
about half the space as character format in this case. Of course, you
can't store just 47 bits exactly in MySQL; you'll probably spend 64
bits of storage space (seems like the next logical boundary, haven't
checked the docs for specifics though). But even so, it cheaper than
storing numbers as characters.
And of course, this is probably a trivial matter unless you have huge
amounts of numbers to store.