For each odd prime p, the p common differences of the arithmetic progressions mentioned in S001075.
0, 2, 2, 0, 2, 6, 6, 6, 0, 4, 6, 12, 30, 30, 150, 0, 2, 6, 6, 30, 60, 123480, 1210230, 32671170, 224494620, 1536160080, 0, 4, 24, 30, 90, 90, 244230, 2190090, 111739740, 111739740, 19458564510
1
As can be seen in the plot, the common differences grow rapidly.
T. D. Noe, Plot of 37 terms
Wikipedia, Primes in arithmetic progression
(Mma) nn = 4; t = {}; Do[p = Prime[n]; s = {p}; k = 0; While[Length[s] < p, k = k + 2; p2 = p; prod = p; cnt = 1; While[p2 = p2 + k; PrimeQ[p2], cnt++; prod = prod*p2; If[cnt > Length[s], AppendTo[s, k]]]]; s[[1]] = 0; AppendTo[t, s], {n, 2, nn}]; t
Cf. S001075.
nonn,tabf,hard
T. D. Noe, Nov 01 2017