# Primes of the form $\frac{n^2-n+4}{2}$ satisfy Hardy-Littlewood analogue?

Let $$n,a,b$$ be positive integers with $$a. Consider primes of the form $$f(n)=\dfrac{n^2-n+4}{2}$$. Let $$C(a,b)$$ denote the amount of primes of the form $$f(n)$$ between (and including) $$f(a)$$ and $$f(b)$$. Now it appears that $$C(a,b) < C(1,b-a+1)+1$$. This observation is an analogue of the second Hardy-Littlewood conjecture.( http://en.wikipedia.org/wiki/Second_Hardy%E2%80%93Littlewood_conjecture)

If we assume there are an infinite amount of primes of the form $$f(n)$$ and/or the original second Hardy-Littlewood conjecture is true , can we prove that $$C(a,b) < C(1,b-a+1)+1$$?

Is it possible to say anything about $$C(a,b) < C(1,b-a+1)+1$$ without assuming an infinite amount of primes of the form $$f(n)$$ and/or the original second Hardy-Littlewood conjecture is true ?

Is there a counterexample known ?

Notice that a finite amount of primes $$f(n)$$ is also potentially consistant with $$C(a,b) < C(1,b-a+1)+1$$.

I was thinking about using a weaker form of the second Hardy-Littlewood namely $$\pi(2,x+1)+1>\pi(y,y+x)$$ where $$\pi(x,y)$$ means counting primes from $$x$$ till $$y$$. But with no success so far.