sage: prod( primes(15) )
30030
sage: _.factor()
2 * 3 * 5 * 7 * 11 * 13

N = 16
R = [1..N]
bad_n_values = []
matches = {}    # dictionary n -> j such that n+j shares no prime factors

for n in [0..prod(primes(N))-1]:

    matches[n] = None    # so far
    for j in R:
        k_match_was_found = False    # so far
        for k in R:
            if k == j:    continue
            if gcd(n+j, n+k) > 1:
                k_match_was_found = True
                break    # the k loop
        if not k_match_was_found:
            matches[n] = j
            break    # the j loop
    # if this point is reached, then matches[n] remains None

if None in matches.values():
    print "NO MATCHES for the following values of n:"
    print [ j for j in matches.keys() if matches[j] == None ]

else:
    print "OK"

OK

sage: for n in [1..20]: print n, '->', n+matches[n]
1 -> 11
2 -> 11
3 -> 11
4 -> 11
5 -> 11
6 -> 13
7 -> 13
8 -> 13
9 -> 13
10 -> 17
11 -> 17
12 -> 17
13 -> 17
14 -> 17
15 -> 17
16 -> 17
17 -> 19
18 -> 19
19 -> 23
20 -> 23

P = prod(list(primes(15)))    # 30030
L = [ n for n in [17..30030.next_prime()] if gcd(n, P) == 1 ]

gaps_list = [ L[k+1] - L[k] for k in range(len(L)-1) ]
maximal_gap = max( gaps_list )
print "Maximal gap is", maximal_gap
print "Following intervals have a gap >16:"
for k in range(len(L)-1):
    if L[k+1] - L[k] > 16:
        print L[k], L[k+1], ':: gap =',  L[k+1] - L[k]

Maximal gap is 22
Following intervals have a gap >16:
2183 2201 :: gap = 18
5749 5767 :: gap = 18
9109 9127 :: gap = 18
9439 9461 :: gap = 22
13339 13357 :: gap = 18
16673 16691 :: gap = 18
20569 20591 :: gap = 22
20903 20921 :: gap = 18
24263 24281 :: gap = 18
27829 27847 :: gap = 18

for k in range(len(L)-1):
    if L[k+1] - L[k] > 16:
        print "INTERVAL (", L[k], ',', L[k+1], ' ) :: gap =',  L[k+1] - L[k]
        for n in range( L[k]+1, L[k+1]-16 ):
            solution = n + matches[n]
            print "tn = %s -> %s = %s" % ( n, solution, solution.factor() )

INTERVAL ( 2183 , 2201  ) :: gap = 18
        n = 2184 -> 2197 = 13^3
INTERVAL ( 5749 , 5767  ) :: gap = 18
        n = 5750 -> 5759 = 13 * 443
INTERVAL ( 9109 , 9127  ) :: gap = 18
        n = 9110 -> 9119 = 11 * 829
INTERVAL ( 9439 , 9461  ) :: gap = 22
        n = 9440 -> 9449 = 11 * 859
        n = 9441 -> 9449 = 11 * 859
        n = 9442 -> 9449 = 11 * 859
        n = 9443 -> 9449 = 11 * 859
        n = 9444 -> 9451 = 13 * 727
INTERVAL ( 13339 , 13357  ) :: gap = 18
        n = 13340 -> 13351 = 13^2 * 79
INTERVAL ( 16673 , 16691  ) :: gap = 18
        n = 16674 -> 16679 = 13 * 1283
INTERVAL ( 20569 , 20591  ) :: gap = 22
        n = 20570 -> 20579 = 13 * 1583
        n = 20571 -> 20579 = 13 * 1583
        n = 20572 -> 20579 = 13 * 1583
        n = 20573 -> 20579 = 13 * 1583
        n = 20574 -> 20579 = 13 * 1583
INTERVAL ( 20903 , 20921  ) :: gap = 18
        n = 20904 -> 20911 = 11 * 1901
INTERVAL ( 24263 , 24281  ) :: gap = 18
        n = 24264 -> 24271 = 13 * 1867
INTERVAL ( 27829 , 27847  ) :: gap = 18
        n = 27830 -> 27841 = 11 * 2531