[math-fun] odd prime query

A friend would like to know if someone can prove or disprove the following assertion: Let d(n) be the characteristic function of the odd primes, p(1)=3, p(2)=5, etc. For j>=3, define S(k,j)=Product{1-d(2j-p(i)): i=1,2,...,k}} if 2j>p(k) and S(k,j)= 0 otherwise. Let A(k,n)=Sum{S(k,j): j=3,4,...,n}, for n>=3. Assertion: A(k,n) < n(1-2/log(2n))^k. Have fun! Clark Kimberling