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- Luke Rhinehart August 17, 2016# A fraction p/q is irreducible and smaller equal 1 if gcd(p,q)=1 and p < q. # Hence the number of irreducible functions of the form i/j where i < j equals phi(j) # where phi is the Euler's totient function, that is summing phi(j) over j= 2,..,n # (n being the input number) we get the desired result. # First get prime numbers smaller or equal than input number n via Eratostene's sieve # and store them in an array O(nlog(log(n))). # For every i=m,...,n compute Euler's totient function phi(i) via the product formula using # the array of primes to find the prime factors of i. Finally sum all phi(i). def count_irr_2(m,n) : primes = eratostene(n) nr = 0 for i in range (m, n+1) : k = 2 phi_i = i check = i while k <= i : if primes[k-2] and i%k == 0 : phi_i = phi_i*(1-1/k) check = int(i/k) elif primes[k-2] and k > check: break k += 1 nr += phi_i return int(nr) def eratostene(n): num = [True]*(n-1) i = 2 while(i*i <= n): if num[i-2] == True : j = i*i-2 while( j < n-1 ): num[j] = False j = j + i i += 1 return num