Interesting question

[David Wilson]

Let S(n) be the set of positive integers not of the form 
x1*x2+x2*x3+x3*x4+...+x_n*x1 for positive integers x1...x_n.

Here is my conjecture:

S(1) = nonsquares (provable).
S(2) = odds (provable).
S(3) = {1,2,4,6,10,18,22,30,42,58,70,78,102,130,190,210,330,462} = 
A025052 (conjectural).
S(4) = primes U {1} (provable).
S(5) = {1,2,3,4,6,8,12,14,18,30,38,42,62}= A101902 (conjectural).
S(6) = {1,2,3,4,5,7,9,13,15,19,21,31,39} (conjectural).
S(n) = {1,2,...,n-1} U {n+1,n+3,n+7,n+9,n+13,n+15}, n >= 7 (conjectural).