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

Department of Mathematics

Indian Institute of Technology

Hauz Khas

New Delhi – 110016

India

**Abstract:**

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*).

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

Return to