In 1939, Jack Chernick proved that, for n ≥ 3 and m ≥ 1:

   U(n, m) = (6m + 1) * (12m + 1) * Product_{i=1..n-2} (2^i * 9m + 1)

is a Carmichael number if all the factors are primes and, for n > 4, m is a multiple of 2^(n-4).



Example
   U(3, m) = (6m + 1) * (12m + 1) * (18m + 1)
   U(4, m) = U(3, m) * (2^2 * 9m + 1)
   U(5, m) = U(4, m) * (2^3 * 9m + 1)
   ...
   U(n, m) = U(n-1, m) * (2^(n-2) * 9m + 1)

 The smallest Chernick's Carmichael number with 3 prime factors, is: U(3, 1) = 1729.
 The smallest Chernick's Carmichael number with 4 prime factors, is: U(4, 1) = 63973.
 The smallest Chernick's Carmichael number with 5 prime factors, is: U(5, 380) = 26641259752490421121.


For n = 5, the smallest number m that satisfy Chernick's conditions, is m = 380, therefore U(5, 380) is the smallest Chernick's Carmichael number with 5 prime factors.

U(5, 380) is a Chernick's Carmichael number because m = 380 is a multiple of 2^(n-4), where n = 5, and the factors { (6*380 + 1), (12*380 + 1), (18*380 + 1), (36*380 + 1), (72*380 + 1) } are all prime numbers.



Task
For n ≥ 3, let a(n) be the smallest Chernick's Carmichael number with n prime factors.

 Compute a(n) for n = 3..9.
 Optional: find a(10).

Note: it's perfectly acceptable to show the terms in factorized form:

 a(3) = 7 * 13 * 19
 a(4) = 7 * 13 * 19 * 37
 a(5) = 2281 * 4561 * 6841 * 13681 * 27361
 ...



See also
 Jack Chernick, On Fermat's simple theorem (PDF)
 OEIS A318646: The least Chernick's "universal form" Carmichael number with n prime factors


 Related tasks
 Carmichael 3 strong pseudoprimes


 
"""
 
Python implementation of
http://rosettacode.org/wiki/Chernick%27s_Carmichael_numbers
 
"""
 
# use sympy for prime test
 
from sympy import isprime
 
# based on C version
 
def primality_pretest(k):
    if not (k % 3) or not (k % 5) or not (k % 7) or not (k % 11) or not(k % 13) or not (k % 17) or not (k % 19) or not (k % 23):
        return (k <= 23)
 
    return True
 
def is_chernick(n, m):
 
    t = 9 * m
 
    if not primality_pretest(6 * m + 1):
        return False
 
    if not primality_pretest(12 * m + 1):
        return False
 
    for i in range(1,n-1):
        if not primality_pretest((t << i) + 1):
            return False
 
    if not isprime(6 * m + 1):
        return False
 
    if not isprime(12 * m + 1):
        return False
 
    for i in range(1,n - 1):
        if not isprime((t << i) + 1):
            return False
 
    return True
 
for n in range(3,10):
 
    if n > 4:
        multiplier = 1 << (n - 4)
    else:
        multiplier = 1
 
    if n > 5:
        multiplier *= 5
 
 
    k = 1
 
    while True:
        m = k * multiplier
 
        if is_chernick(n, m): 
            print("a("+str(n)+") has m = "+str(m))
            break
 
        k += 1
 