 Trial-division divisor counting

Factor a positive integer while multiplying the exponent-plus-one contributions to its divisor count. After removing all prime factors below the current trial factor, a trial factor that divides the remainder must itself be prime. Remove its full power and advance. Once the trial factor's square exceeds the remainder, any remainder greater than one is prime and contributes a factor two. The invariant separates the original integer into removed prime powers and the remaining coprime factor; no factor array is necessary.