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The Frobenius number tells the largest sum that cannot be made from linear combinations of a set of integers, e.g. $43$ for $\{6, 9, 20\}$.

When the gcd of the set of integers is $g$, I am looking for a way to formalize the number $n$ representing the lowest sum past which all possible sums are separated by $g$. For example, for $\{6, 16 \} (g = 2)$ the possible sums are $[0, 6, 12, 16, 18, 22, 24, 28, 30, ...]$. From $28$, the possible sums are $2$ apart, so $n$ is $28$. For $\{12, 20, 44\}$ with $g = 4, n = 32$.

We may assume (without loss of generality, I believe) that the set of integers does not contain numbers that are multiples of one another. e.g. $\{6, 16, 18\} \equiv \{6, 16 \}$.

smichr
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    Your problem is the same as Frobenius problem, if you want the least integer as you said, it is $(F(\dfrac{{A}}{g})+1)g$, $F({A})$ is the Frobenius number of the set ${A}$. – Toni Mhax Nov 11 '19 at 05:40
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    Thank you, Frogenius! Now I see it: factoring out g gives us a set that will represent all sums past its Frobenius number. So add 1 to get the first number of the continuous range. Multiplying by $g$ gives all numbers separated by $g$. – smichr Nov 11 '19 at 05:58

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