Just saw this 1928 theorem discussed on sci.math.research: given any c, 0 < c < 1, the set S(c) of positive integers n satisfying phi(n)/n <= c has a well-defined density D(c) in {1,2,3,...}. And, D(c) is a continuous function of c. It's believed to have an infinite derivative at some points, and a (finite) one-sided derivative at the others, but these things seem to be unproven. --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele