Why is Sieve of Eratosthenes more efficient than the simple “dumb” algorithm?

Question

If you need to generate primes from 1 to N, the \"dumb\" way to do it would be to iterate through all the numbers from 2 to N and check if the numbers are divisable by any p

Answer 1

http://en.wikipedia.org/wiki/Prime_number#Number_of_prime_numbers_below_a_given_number

• the "dumb" algorithm does i/log(i) ~= N/log(N) work for each prime number
• the real algorithm does N/i ~= 1 work for each prime number

Multiply by roughly N/log(N) prime numbers.