Compute OEIS sequence A144311; see http://math.stackexchange.com/questions/1779109.

Question1779109.java

import java.util.ArrayList;
import java.util.List;

public class Question1779109 {
	final static int [] p = {2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97};
	final static int maxLength = 2000;
	final static int ndirect = 7;

	static int max;
	
	public static void main (String [] args) {
		for (int n = ndirect;;n++) {
			recurseDirectly (n,0,new int [n]);
			System.out.println (n + " : " + (max - 1));
		}
	}
	
	static void recurseDirectly (int n,int np,int [] residues) {
		if (np < ndirect) {
			for (residues [np] = 0;residues [np] < p [np] - 1;residues [np]++)
				if (residues [np] != 1)
					recurseDirectly (n,np + 1, residues);
		}
		else {
			List<Integer> remainsList = new ArrayList<Integer> ();
			outer:
			for (int r = 1;r <= maxLength;r++) {
				for (int i = 0;i < ndirect;i++) {
					int res = (r + residues [i]) % p [i];
					if (res == 1 || res == p [i] - 1)
						continue outer;
				}
				remainsList.add (r);
			}
			int [] remains = new int [remainsList.size ()];
			for (int i = 0;i < remains.length;i++)
				remains [i] = remainsList.get (i);
			recurse (n,0,ndirect,remains,new boolean [n],new int [n],residues);
		}
	}
	
	static void recurse (int n,int d,int nused,int [] remains,boolean [] used,int [] primes,int [] residues) {
		outer:
		for (;;) {
			max = Math.max (max,remains [d]);
			for (int i = ndirect;i < nused;i++) {
				int r = (residues [i] + remains [d]) % primes [i];
				if (r == 1 || r == primes [i] - 1) {
					d++;
					continue outer;
				}
			}
			break;
		}
	
		for (int i = ndirect;i < n;i++)
			if (!used [i])
				for (int r = -1;r <= 1;r += 2) {
					int res = p [i] - ((r + remains [d] - 1) % p [i] + 1);
					if (res != 1 && res != p [i] - 1) {
						residues [nused] = res;
						primes [nused] = p [i];
						used [i] = true;
						recurse (n,d + 1,nused + 1,remains,used,primes,residues);
						used [i] = false;
					}
				}
	}
}