Suppose we want to estimate the number of positive integers <= x that are products of odd numbers. Piece of cake — these are just the odd numbers themselves, to x/2 is a good asymptotic estimate. Now suppose that for given N in Z+ and q with 0 <= q < N we want to estimate the number #(x; N, q) of positive integers <= x that are product of numbers of the form kN + q, for any various values of k in Z+. Example: #(150; 5, 4) = |{4x4, 4x9, 4x14, 4x19, 4x24, 4x29, 4x34, 9x9, 9x14, 4x4x4, 4x4x9}| = 11 . Heuristically, for each integer X <= x this should be pretty close to the case where q = 0. Which would imply that if we put the integer x in the form: X = L N^s with gcd(L, N) = 1, and s in Z[>=0], then YIKES, this will already involve the partitions P of s each paired with the number of ways L can be factored into |P| factors — counting order but avoiding duplicates . . . added over all X <= x. There must be a better way. —Dan