Generate *all* coprime tuples Given integers k and n, generate a sequence of n unique k-tuples of pairwise coprime integers. Every such tuple must occur once eventually, that is, for any existing k-tuple of pairwise coprime integers, some n will eventually generate it.
The output may be printed or evaluated in any list/tuple-like form.
Definitions

Two numbers a and b are coprime if gcd(a, b) = 1, i.e. they share no common divisor other than 1.
A tuple of k numbers (a1, a2, ..., ak) is pairwise coprime if every pair of numbers in the tuple is coprime.

Examples
 k =  1, n =  5 -&gt; [[1],[2],[3],[4],[5]]
 k =  2, n =  7 -&gt; [[2,1],[3,1],[3,2],[4,1],[4,3],[5,1],[5,2]]
 k =  3, n = 10 -&gt; [[3,2,1],[4,3,1],[5,2,1],[5,3,1],[5,3,2],[5,4,1],[5,4,3],[6,5,1],[7,2,1],[7,3,1]]
 k =  4, n =  2 -&gt; [[5,3,2,1],[5,4,3,1]]
 k =  5, n =  0 -&gt; []

Notes

Standard code golf rules, shortest code wins.
k is assumed to be positive, and n non-negative.
The numbers within each tuple must be positive, distinct, and may appear in any order.
Uniqueness is up to ordering: e.g. (1,2,3) is the same as (1,3,2).
Good luck and have fun!
05AB1E, 13 bytes

I think it has been 389 days since I last posted something here haha. There is definitely some golfing potential left in this program.

Code

Uses the 05AB1E-encoding.

∞æ¹ùʒPy.¿Q}²£


Try it online!



Explanation

It is worth noting that for two numbers \$n, m \in \mathbb{Z}^+\$ that:

$$
\tag{1} \label{1}
\gcd(n, m) \cdot \text{lcm}(n, m) = n \cdot m
$$

This means that for two numbers \$n, m \in \mathbb{Z}^+\$ where the \$\gcd(n, m) = 1\$, we can conclude that the \$\text{lcm}(n, m) = n \cdot m\$.

Furthermore, the \$\gcd\$ function is a multiplicative function, which means that if \$n_1\$ and \$n_2\$ are relatively prime, then:

$$
\gcd(n_1 \cdot n_2, m) = \gcd(n_1, m) \cdot \gcd(n_2, m)
$$



From this, we obtain the fact that:

$$
\tag{2} \label{2}
\gcd(a, bc) = 1 \iff \gcd(a, b) = 1 \wedge \gcd(a, c) = 1
$$



Let us denote a \$k\$-tuple of positive integers as \$S = \{x_1, x_2, \dots, x_k\}\$. A set \$S\$ is pairwise coprime, if and only if:  

$$
\tag{3} \label{3}
\forall a, b \in S \wedge a \not = b \rightarrow \gcd(a, b) = 1
$$



Using Equations \$\eqref{1}, \eqref{2}\$ and \$\eqref{3}\$, we can conclude that a set \$S = \{x_1, x_2, \dots, x_k\}\$ is pairwise coprime, if and only if:

$$
\text{lcm}(x_1, x_2, \dots, x_k) = \prod_{x \in S} x
$$

Code Explanation


∞æ¹ùʒPy.¿Q}²£

∞æ              # Powerset of the infinite list [1, ..., ∞].
  ¹ù            # Keep only lists of length k.
    ʒ     }     # Filter. Keep lists where the
     P          #   product of the list
         Q      #   is equal to
      y.¿       #   the least common multiple of the list
           ²£   # Retrieve the first n elements.