Developments 

This decomposition is always possible except for p(1) = 2, p(2) = 3 and p(4) = 7 or p(n+1) >= (3/2) * p(n) holds only for n = 1, n = 2 and n = 4 (for the proof see arXiv:0711.0865). 

p is a prime such that p > 3 and p + 2 is also prime if and only if p has a weight equal to 3. (see arXiv:0711.0865). 

For p(n) different from 2, 3 and 7, we have : 
l(n) = p(n)  g(n) = 2 * p(n)  p(n+1) = k(n) * L(n), 
p(n) = l(n) + g(n) = k(n) * L(n) + g(n), 
gcd(g(n),2) = 2, 
gcd(p(n),g(n)) = gcd(p(n)  g(n),g(n)) = gcd(l(n),g(n)) = gcd(L(n),g(n)) = gcd(k(n),g(n)) = 1, 
3 <= k(n) <= l(n), 
1 <= L(n) <= l(n) / 3, 
2 <= g(n) <= k(n)  1, 
2 * g(n) + 1 <= p(n). 

