Partial Sums of Powers of Prime Factors

Given integers k >= 2 and l >= 3, let S-k,l(*) stand for the of those positive integers n which can be written as n= p(1)(k) + p(2)(k) + ...+ p(l)(k) , where p(1) , p(2) ,... , p(l) are distinct prime factors of n. We study the properties of the sets S-k,l(*) and we show in particular that given any odd l >= 3, # boolean OR(infinity)(k=2) S-k,l(*) = +infinity.