Sums of numbers with many divisors

Let k be a fixed integer, k greater than or equal to 2, and suppose that ep silon>0. We show that every sufficiently large integer n can be expressed i n the form n = m(1) + m(2) +...+ m(k) where d(m(i)) > n((log2-epsilon)(1-1/ k)/log log n) for all i. This is best possible, since there are infinitely many exceptional n if the factor log 2 - epsilon is replaced by log 2 + eps ilon. (C) 1999 Academic Press.