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

Is this from Project Euler 245? I remember that question, and I have given up on it.

The fastest way I found for calculating totient was to multiply the prime factors (p-1) together, given that k has no repeated factors (which was never the case if I remember the problem correctly).

So for calculating factors, it would probably be best to use gmpy.next_prime or pyecm (elliptic curve factorization).

You could also sieve the prime factors as Jaime suggests. For numbers up to 10¹², the maximum prime factor is below 1 million which should be reasonable.

If you memoize factorizations, it could speed up your phi function even more.