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

God created the natural numbers. All the rest is the work of man. Abstract The paper considers the problem of finding the largest possible set P(n), a subset of the set N of the natural numbers, with the property that a number is in P(n) if and only if it is a sum of n distinct naturals all in P(n) or none in P(n). Here " largest " is in the set theoretic sense and n≥2. We call P(n) a maximal set obeying this property. For small n say 2 or 3, it is possible to develop P(n) intuitively but we strongly felt the necessity of an algorithm for any n≥2. Now P(n) shall invariably be a infinite set so we define another set Q(n) such that Q(n)=N-P(n), prove that Q(n) is finite and, since P(n) is automatically known if Q(n) is known, design an algorithm of worst case O(1) complexity which generates Q(n). The paper is organized as follows. Section 1 is the introduction where the algorithm is designed after a brief statement of the problem. Section 2 gives the code. Section 3 gives several examples with outputs of the code provided in the previous section. Finally section 4 is the conclusion and future work.