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
Radical of an integer no known polynomial-time algorithm for computing the square-free part of an integer. The definition is generalized to the largest "t"-free divisor of "n", formula_5, which are multiplicative functions which act on prime powers as The cases "t"=3 and "t"=4 are tabulated in and . The notion of the radical occurs in the abc conjecture, which states that, for any "ε" > 0, there exists a finite "K" such that, for all triples of coprime positive integers "a", "b", and "c" satisfying "a" + "b" = "c", For any integer formula_8, the nilpotent elements of the finite ring formula_9 are all