Finding the numbers $n$ such that $2n+5$ is composite. Let $n$ be a positive integer greater than zero. I write 

$$a_n =
\begin{cases}
1 , &amp;\text{ if } n=0 \\
1 , &amp;\text{ if } n=1 \\
n(n-1), &amp; \text{ if $2n-1$ is prime} \\
3-n, &amp; \text{ otherwise}
\end{cases}$$

The sequence goes like this $$1,1,2,6,12,-2,30,42,-5,72,90,-8,132,-10,-11,\ldots$$ I would like to prove the following two claims.


  claim 1 : If $a_n&gt;0$ and
  ${a_n \above 1.5 pt 3} \notin \mathbb{Q}$ then $\sqrt{4a_n+1}$ is prime. 


The table below illustrates what I am seeing: 

\begin{array}{| l | l | l | l }
\hline
n &amp; a_n  &amp; {a_n \above 1.5 pt 3} &amp; \sqrt{4a_n+1}\\ \hline
0 &amp; 1 &amp;  .333333.. &amp; 2.2360679.. \\ 
1 &amp; 1 &amp;  .333333.. &amp; 3 \\ 
2 &amp; 2 &amp;  .666666..  &amp; 3   \\ 
3 &amp; 6 &amp;  2   &amp; 5   \\ 
4 &amp; 12 &amp;  4   &amp; 7   \\ 
6 &amp; 30 &amp; 10   &amp; 11    \\ 
7 &amp; 42 &amp; 14  &amp; 13     \\ 
9 &amp; 72 &amp; 24  &amp; 17 &amp;   \\ 
10 &amp; 90 &amp; 30   &amp; 19  \\ 
12 &amp; 132 &amp; 44   &amp; 23  \\ 
15 &amp; 210 &amp; 70   &amp; 29  \\ 
16 &amp; 240 &amp; 80  &amp; 31  \\ 
19 &amp; 342 &amp; 114   &amp; 37 \\ 
21 &amp; 420 &amp; 140   &amp; 41 \\ 
22 &amp; 462 &amp; 154   &amp; 43 \\ 
    \hline
    \end{array}


  claim 2: If $a_n&lt;0$ then $2a_n+5$ is composite
$n=13$ is a counter example for claim 2.
Claim 1 is vaccuously true, since given any $n$, $a_{n}\in\mathbb{Z}$.

It also looks like n=14 is a counterexample to claim 2.

Proof: 
$3-14=-11$ and $2(-11)+5=-17$ has positive divisors $1$ and $17$ only.