TITLE: Generating all coprime pairs within limits
QUESTION [13 upvotes]: Say I want to generate all coprime pairs ($a,b$) where no $a$ exceeds $A$ and no $b$ exceeds $B$.
Is there an efficient way to do this?

REPLY [3 votes]: From the wikipedia page for "Coprime integers"   (https://en.wikipedia.org/wiki/Coprime_integers):
Generating all coprime pairs
All pairs of positive coprime numbers $(m,n)$ (with $m>n$) can be arranged in two disjoint complete ternary trees, one tree starting from $(2,1)$ (for even-odd and odd-even pairs), and the other tree starting from $(3,1)$ (for odd-odd pairs). The children of each vertex $(m,n)$ are generated as follows:

Branch 1: $(2m-n,m)$
Branch 2: $(2m+n,m)$
Branch 3: $(m+2n,n)$

This scheme is exhaustive and non-redundant with no invalid members.
For programming purposes:
Knowing that every child has greater $m$ than its parent, the tree can be generated depth first following, at each node, the first available branch that fulfills $m<A$ until all three are exhausted.
When a node has exhausted its three branches, the parent node can be generated without any extra information knowing the branch used to generate the current node:
if $((2n-m)≥0)$, branch 1
else if $(3n≥m)$, branch 2
else, branch 3
with branch and current node, generate parent node and then follow the next available branch to the one already explored.
Following this method, it is possible to generate both trees $(2,1)$ and $(3,1)$ starting with these two nodes and ending when $(1,0)$ and $(1,1)$ are reached respectively.
This allows the programming of a function with inputs the current node and the limit and outputs the next node in the tree.