A combinatorial problem that arose in integer B3 Sets

Let A = {a1, a2, …, ak} be a set of positive integers with k ≥ 3,such that a1 ≤ a2 ≤ a3 … ak = N. Our problem is to investigate thenumber of triplets (ar, as, at) ∈ A3 with ar < as < at, satisfyingar + as - at < 0 y -ar + as + at > N.In this paper we give an upper bound for the maximum number of such a triplets in an arbitrary set of integers with k elements. We also find the number of triplets satisfying () for some families of sets in order to determine lower bounds for the maximum number of such a triplets that a set with k elements can have.