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The paper considers the problem of finding the largest possible set P(n), asubset of the set N of the natural numbers, with the property that a number isin P(n) if and only if it is a sum of n distinct naturals all in P(n) or nonein P(n). Here largest is in the set theoretic sense and n is greater than orequal to 2. We call P(n) a maximal set obeying this property. For small n say 2or 3, it is possible to develop P(n) intuitively but we strongly felt thenecessity of an algorithm for any n greater than or equal to 2. Now P(n) shallinvariably be a infinite set so we define another set Q(n) such thatQ(n)=N-P(n), prove that Q(n) is finite and, since P(n) is automatically knownif Q(n) is known, design an algorithm of worst case O(1) complexity whichgenerates Q(n).

On an algorithm that generates an interesting maximal set P(n) of the naturals for any n greater than or equal to 2