Counting sort initialization of the output array, each take time. Therefore, the time for the whole algorithm is the sum of the times for these steps, . Because it uses arrays of length and , the total space usage of the algorithm is also . For problem instances in which the maximum key value is significantly smaller than the number of items, counting sort can be highly space-efficient, as the only storage it uses other than its input and output arrays is the Count array which uses space . If each item to be sorted is itself an integer, and used as
Square-free integer binary digits are reversed from the ordering in the infinite product.) Since the prime factorization of every number is unique, so also is every binary encoding of the square-free integers. The converse is also true. Since every positive integer has a unique binary representation it is possible to reverse this encoding so that they may be decoded into a unique square-free integer. Again, for example, if we begin with the number 42, this time as simply a positive integer, we have its binary representation 101010. This decodes to 2 · 3 · 5 · 7 · 11 · 13 =