Denominators in the Farey Sequence, the Stern Sequence, and the Frobenius Problem with Relatively Prime Multipliers

The Frobenius problem asks, given two positive, relatively prime integers $a$ and $b$, what is the maximum integer that cannot be expressed as a linear combination of $a,b$ using non-negative coefficients? This problem is resolved. However, if we add the restriction that the coefficients themselves must also be relatively prime, we create an open problem with connections to the Stern and Farey sequences and the Jacobsthal function. We prove that under this new restriction, a maximum inexpressible integer still always exists, and explore its values and properties as functions of $a,b$.