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


#include "isprime.bas"
 
Function PrimalityPretest(k As Integer) As Boolean
    Dim As Integer ppp(1 To 8) = {3,5,7,11,13,17,19,23}
    For i As Integer = 1 To Ubound(ppp)
        If k Mod ppp(i) = 0 Then Return (k <= 23)
    Next i
    Return True
End Function
 
Function isChernick(n As Integer, m As Integer) As Boolean
    Dim As Integer i, t = 9 * m
 
    If Not PrimalityPretest(6 * m + 1) Then Return False
    If Not PrimalityPretest(12 * m + 1) Then Return False
 
    For i = 1 To n-1
        If Not PrimalityPretest(t * (2 ^ i) + 1) Then Return False
    Next i
 
    If Not isPrime(6 * m + 1) Then Return False
    If Not isPrime(12 * m + 1) Then Return False
 
    For i = 1 To n - 2
        If Not isPrime(t * (2 ^ i) + 1) Then Return False
    Next i
    Return True
End Function
 
Dim As Uinteger multiplier, k, m = 1
For n As Integer = 3 To 9
    multiplier = Iif (n > 4, 2 ^ (n-4), 1)
    If n > 5 Then multiplier *= 5
 
    k = 1
    Do
        m = k * multiplier
        If isChernick(n, m) Then
            Print "a(" & n & ") has m = " & m
            Exit Do
        End If
        k += 1
    Loop
Next n
Sleep