I am given a set of numbers $(n_1, n_2, ...)$ with $n_1 < n_2 < ...$ and I want to know what the largest number is that can't be written as $a_1*n_1 + a_2*n_2 + ...$ The set of numbers is always finite and all $n_i$ and $a_i$ are positive integers. Is there a general technique to do this, or do you just have to try until you get $n_1$ consecutive numbers?
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Example: n is (3,5), largest number is 7 – LeeNeverGup Nov 04 '13 at 13:12
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Indeed, since $n_1=3$ and 8, 9 and 10 can be formed, each number larger than 8 can be formed. – bart_m8 Nov 04 '13 at 13:16
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But if n is (10, 100) any number not divisable by 10 will fit. The LCS of the ns must be 1. – LeeNeverGup Nov 04 '13 at 13:18
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What do you mean by LCS? – bart_m8 Nov 04 '13 at 13:24
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I meant gcd, the greatets common divisor – LeeNeverGup Nov 04 '13 at 13:26
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Uhu, then I follow you. I thought LCS might be some variation on LCM. But OK, you are right the GCD needs to be 1. – bart_m8 Nov 04 '13 at 13:27
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1It's the Frobenius coin problem. – bof Nov 04 '13 at 13:33
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Thanks. I hadn't heard of that name. It is indeed what I'm looking for. – bart_m8 Nov 04 '13 at 13:35
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We want to assume, as the Comments note, that the GCD of values $n_i$ is 1, as otherwise there is no upper bound on numbers that cannot be represented.
Solving this, the Frobenius coin problem, has previously been discussed here.
Edit: Technically this Question differs from the Coin Problem by requiring the count $a_i$ of each denomination $n_i$ to be "positive", where these coefficients are allowed to be zero in the Coin Problem. Whether or not this was a miswording of the Question, the only effect is to add $n_1 + n_2 + \ldots$ to each possible sum, so the set of representable values is simply shifted upward by this amount.
For general cardinality of $\{n_i\}$, stipulated to be finite, the problem is NP-hard as noted in an on-line paper by D. Beihoffer et al that discusses speed of algorithms. However for fixed number of integers $n_i > 0$ there are algorithms of polynomial complexity in size of input (logarithms of $n_i$'s).
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1See also https://brg.a2hosted.com/?page_id=563 for a Python implementation of
frobenius_number. – smichr Nov 11 '19 at 10:07