Ratio of square numbers to numbers with 4 factors?

Answer 1

As N approaches infinity the ratio of squares less than N to numbers with 4 factors less than N approaches 0. This means that in the customary way of defining it, the ratio you're interested in is 0 (although that should be taken with a grain of salt - it certainly doesn't mean that there are 0 square numbers). The number of squares less than N is approximately √N. Rather than calculating the ratio we're interested in, we're going to calculate a calculate a ratio guaranteed to be greater: the ratio of squares to numbers that are twice a Prime number (which are some, but not all, of the numbers with 4 factors). There are approximately N/ln N prime numbers less than N, by the prime number theorem. So there are N/(2 ln N/2) prime numbers less than N/2, which can be doubled to get a number less than N that's twice a prime number. The ratio is therefore √N(2 ln N/2)/N, which is O(ln N/√N). √N grows much faster than ln N, and in the limit this ratio will get close to zero. So the ratio we're actually interested in, which is even less than this ratio, will also approach zero.