Conjectured value of k for numbers in S000064.
1, 1, 1, 2, 1, 1, 2, 1, 1, 9, 1, 17, 2, 1, 1, 75, 892, 5, 74, 1, 867709, 517413, 1, 6628, 4352, 1, 1, 44, 1, 404142
2
T. D. Noe, Plot of terms 2 to 31
(Mma) Table[mx = {}; lastI = 0; Do[s = Select[n*i + Range[n], PrimeQ]; If[Length[s] > Length[mx], lastI = i; mx = s - n*i], {i, 10^7}]; lastI, {n, 2, 31}]
nonn,hard
T. D. Noe, May 29 2014