Journal of Integer Sequences, Vol. 20 (2017), Article 17.7.2

On a Special Case of the Frobenius Problem

Amitabha Tripathi
Department of Mathematics
Indian Institute of Technology
Hauz Khas
New Delhi – 110016


For any set of positive and relatively prime integers A, the set of positive integers that are not representable as a nonnegative integral linear combination of elements of A is always a non-empty finite set. Thus we may define g(A), n(A), s(A) to denote the largest integer in, the number of integers in, and the sum of integers in this finite set, respectively. We determine g(A), n(A), s(A) when A = {a, b, c} with a | lcm(b, c). A particular case of this is when A = {kl, lm, mk}, with k, l, m pairwise coprime. We also solve a related problem when a | lcm(b, c), thereby providing another proof of the formula for g(A).

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Received April 27 2017; revised versions May 1 2017; June 15 2017; June 26 2017. Published in Journal of Integer Sequences, July 2 2017.

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