
COMMENTS

Minimal values of s are: 1, 10, 23, 67, 211, 163, 157, 163.
The chains are:
1
25, 27
115, 117, 119
1001, 1003, 1005, 1007
4429, 4431, 4433, 4437, 4439
7657, 7659, 7661, 7663, 7665, 7667
34365, 34371, 34375, 34383, 34385, 34391, 34397
49375, 49377, 49379, 49381, 49383, 49385, 49387, 49389


EXAMPLE

The chain for n = 2 is smallest for s = 10 to s = 12.
The chain for n = 3 is smallest for s = 23 to s = 28.
The chain for n = 4 is smallest for s = 67 to s = 166.
The chain for n = 5 is smallest for s = 211 to s = 276.
The chain for n = 6 is smallest for s = 163 to s = 382.
The chain for n = 7 is smallest for s = 157 to s = 178.
The chain for n = 8 is smallest for s = 163 to s = 822.
Note that for n > 2, the range of s starts with a prime number and ends just before a prime number.


MATHEMATICA

n = 4; s = 1; mn = Infinity; While[s < mn, s++; f = s!; consec = 0; lst = {}; i = 1; found = False; While[If[Mod[f, i] == 0, If[OddQ[i], consec++; AppendTo[lst, i]; If[consec == n, found = True], consec = 0; lst = {}]]; ! found && i < mn && i^n < f, i++]; If[found, Print[{s, lst}]; mn = Min[mn, i]]] (* T. D. Noe, Oct 31 2012 *)
