Is there a known algorithm for computing the n-th Turing machine directly? Let us define a Turing machine by a machine description that is a string of symbols produced by some numerical encoding. For example, a Turing machine $M_1$ can be represented by 9,900,599 ([0 0 halt], where halt = 5 and the start and end of an instruction is 99). A standard enumeration consists of all possible Turing machines.

If given a number $n$, is there a known algorithm out there for producing the $n$th Turing machine ($M_n$) in the standard enumeration?
Generate all strings in order, and keep track of which of them correspond to valid encodings. One you have seen the $n$th one, output it.

There are probably more efficient algorithms, but it depends on your exact encoding. Standard combinatorial enumeration techniques might suffice.