Short Problem Definition:

Count the semiprime numbers in the given range [a..b]

Link

SemiPrimes

Complexity:

expected worst-case time complexity is O(N*log(log(N))+M);

expected worst-case space complexity is O(N+M)

Execution:

First get all semiprimes from an adaptation of the Sieve of Eratosthenes. Because we will be computing the difference many times a prefix sum is adequate. Get the number of semiprimes up to the point. The index P is decreased by 1 because we want to know all primes that start from P.

Solution:

def sieve(N):
    semi = set()
    sieve = [True]* (N+1)
    sieve[0] = sieve[1] = False

    i = 2
    while (i*i <= N):
        if sieve[i] == True:
            for j in xrange(i*i, N+1, i):
                sieve[j] = False
        i += 1

    i = 2
    while (i*i <= N):
        if sieve[i] == True:
            for j in xrange(i*i, N+1, i):
                if (j % i == 0 and sieve[j/i] == True):
                    semi.add(j)
        i += 1

    return semi

def solution(N, P, Q):

    semi_set = sieve(N)

    prefix = []

    prefix.append(0) # 0
    prefix.append(0) # 1
    prefix.append(0) # 2
    prefix.append(0) # 3
    prefix.append(1) # 4

    for idx in xrange(5, max(Q)+1):
        if idx in semi_set:
            prefix.append(prefix[-1]+1)
        else:
            prefix.append(prefix[-1])

    solution = []

    for idx in xrange(len(Q)):
        solution.append(prefix[Q[idx]] - prefix[P[idx]-1])

    return solution

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