interesting sequence

[Jeffrey Shallit]

Here's an interesting sequence for which I don't know the next
term:

	2, 21, 1310, ?

The sequence (S_n) is defined as follows: 

	S_n = the least positive integer r such that there exists
	      an integer s, 0 <= s < r

			gcd(r-i, s-j) > 1

	      for all integers i, j with 0 <= i, j < n.

Then 
	S_1 = 2    because we can choose r = 2,    s = 0
        S_2 = 21   because we can choose r = 21,   s = 15
	S_3 = 1310 because we can choose r = 1310, s = 1276

By brute force search I know that S_4 > 410,000.  And also
I know by constructing the pair (r, s) = (477742707, 172379781)
that S_4 <= 477742707.

It's also possible to give asymptotic estimates on the growth
rate of S_n.  For example, I can show that

	S_n < e^{(1+o(1)) 2 n^2 log n}

The sequence S_n is also related to Jacobsthal's function g(n).

Jeffrey Shallit