A pattern in counting number of primes

The question arises how can we find the carnality of numbers less than a given natural number. Well to start with, we can use the idea of fundamental theorem of number. It states that any number is either a prime or a product or primes.Once we have a prime factorization the power of each of that prime could either be even or odd. If it happens to be even that means its a square number and if it happens to be an odd than its one more than the even number. So we can group all the even parts as one and the single exponents as other. We call the part where the numbers have only single power as non square part of the number. Now if n is the number its easy to see that the square part never exceeds the square root of the number and the non square part could contain as many as pi(n) number of primes at most. If we have to make all possible numbers than for square part we have a choice of sqrt(n) and for non square part we have a choice of 2^pi(n) numbers at most. Notice that if one use all prime numbers less than n it may likely to overshoot the number itself. Thus a rough bound on the numbers can be obtained by fundamental principle of counting as sqrt(n)*2^pi(n) > n