Calculating phi(k) for 1<k<N

Question

Given a large N, I need to iterate through all phi(k) such that 1 < k < N :

• time-complexity must be O(N logN)
• memory-complexity mus

Answer 1

I think you can go the other way around. Instead of factorizing an integer k to get phi(k), you can attempt to generate all integers from 1 to N from primes and get phi(k) quickly. For example, if P_n is the maximum prime that is less than N, you can generate all integers less than N by

P₁ ^{i ₁} * P₂ ^{i ₂} * ... * P_n ^{i _n} where each i_j run from 0 to [log (N) / log (P_j)] ([] is the floor function).

That way, you can get phi(k) quickly wihout expensive prime factorization and still iterate through all k between 1 and N (not in order but I think you don't care about order).