Some Number Theoretic Results

This note consists of two rather separate topics . In §1 we generalize a topic from combinatorial number theory to get an order of magnitude for the number of elements in a maximal set of residues (mod p) such that sums of different numbers of elements from this set are distinct. We show that the correct order is ep'" 3 although we are unable to establish the correct value for the constant c . Section 2 consists of irrationality results on series of the form f (n)/a 1a 2 • • • a , where f(n) is one of the number theoretic functions d(n), u(n) or p(n) and a n are integers >_ 2 . For f(n) = d(n) it suffices that the a n are monotonic while for u(n) and p(n) we needed additional conditions on their rates of growth .