Let A be a subset of N. We say that A is m-full if PA = [m] for a positive integer m, where PA is the set of all positive integers which are a sum of distinct elements of A and [m] = {1, . . . ,m}. In this paper, we show that a set A = {a1, . . . , ak} with a1 <


< ak is full if and only if a1 = 1 and ai ≤ a1+


+ai−1+1 for each i, 2 ≤ i ≤ k. We also prove that for each positive integer m / ∈ {2, 4, 5, 8, 9} there is an m-full set. We determine the numbers α(m) = min{|A| : PA = [m]}, β(m) = max{|A| : PA = [m]},L(m) = min{maxA : PA = [m]} and U(m) = max{maxA : PA = [m]} in terms of m. We also give a formula for F(m), the number of m-full sets.

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