Primes stuff.

output.txt

M_1 = [4, 8, 9, 12, 16, 18, 20, 24, 25, 27]
M_2 = [60, 66, 90, 600, 606, 609, 666, 680, 696, 806]
M_3 = [4, 6, 8, 9, 12, 16, 18, 20, 24, 25]
M_a = [4, 8, 9, 12, 16]
M_b = [6, 60, 66, 69, 86]
M_c = [6, 60, 66, 69, 86]

sets.py

#!/usr/bin/env python2

primes = [2]
for p in xrange(3, 1000, 2):
    prime = True
    q = 3
    while q*q <= p:
        if p % q == 0:
            prime = False
            break
        q += 1
    if prime:
        primes.append(p)

def prime_factors(x):
    if x == 0:
        return []
    prime_factors = []
    primes_i = 0
    rest = x
    while rest != 1:
        if primes_i >= len(primes):
            raise Exception("Not enough prime numbers available!")
        prime = primes[primes_i]
        if rest % prime == 0:
            rest = rest / prime
            prime_factors.append(prime)
        else:
            primes_i += 1
    return prime_factors

def P(x):
    return len(set(prime_factors(x))) == 3

def Q(x):
    if x == 0:
        return False
    for p in primes:
        if x % (p*p) == 0:
            return True
        if p*p >= x*x:
            break
    return False

def R(x):
    for c in str(x):
        if not c in ("0", "6", "8", "9"):
            return False
    return True

if __name__ == "__main__":
    max_num = 1000

    set_negate = lambda X: set(range(1, max_num)).difference(X)
    first_elements = lambda X, n=10: sorted(list(X))[:n]

    P_set, Q_set, R_set = set(), set(), set()
    for x in xrange(1, max_num):
        if P(x):
            #print "P(%s)" % x
            P_set.add(x)
        if Q(x):
            #print "Q(%s)" % x
            Q_set.add(x)
        if R(x):
            #print "R(%s)" % x
            R_set.add(x)

    M_1 = set_negate(P_set).intersection(Q_set)
    M_2 = P_set.intersection(R_set)
    M_3 = set_negate(P_set).intersection(Q_set).union(R_set)

    a = M_1.union(M_2)
    b = M_3.intersection(set_negate(M_1))
    c = M_3.difference(M_1)

    print "M_1 =", first_elements(M_1)
    print "M_2 =", first_elements(M_2)
    print "M_3 =", first_elements(M_3)
    print "M_a =", first_elements(a, 5)
    print "M_b =", first_elements(b, 5)
    print "M_c =", first_elements(c, 5)