Interesting mapping between Q and Z^∞

I found a way to convert between a rational and countably infinitely dimensional lattice a few years ago, and I recently was reminded of it again, I'm guessing it's a "canonical" and "obvious" mapping, but I'll describe it anyways just in case.

Take a positive rational a/b that is fully reduced and factor both the numerator and denominator into prime powers
2^m_1, 3^m_2, 5^m_3, 7^m_4, 11^m_5, ... and 2^n_1, 3^n_2, 5^n_3, 7^n_4, 11^n_5, ...

Observe that if m_i is non-zero, then n_i is 0 and vice versa. This is due to the assumption that a/b is fully reduced, i.e. gcd(a,b) = 1. Also notice that their exists a final non-zero term in both m and n, this is because the rationals don't contain an infinite element; only arbitrarily large, finite elements.

Now create a countably infinite dimensional vector v.
for every positive integer i,
v_i = m_i if m_i =/= 0, else set v_i = n_i if n_i =/= 0, otherwise set v_i = 0

I claim that every point in Z^∞ is able to be hit by a specific value a/b through this conversion to v.

from my definition of v, every dimension in Z^∞ corresponds to a unique prime number, because there is no last prime (Euclid 300BC), we have half the problem down, to show that a point can wander as far away as it wants, we can use the reverse process to find a/b from v.