Properties of $n$ such that $4n+1$ is prime?

Question

I would like to find some properties of natural numbers $n$ such that $4n+1$ is prime, i.e. the sequence $$ 1, 3, 4, 7, 9, 10, 13, 15, 18, 22, 24,... $$ In oeis.org website there is only one relation between this sequence and triangular numbers, that it was also vague.

Is there any other sequences that we can get the above sequence by some operations on them. Any interesting properties of them are welcome and appreciated.

I guess necessary conditions is also useful if any.

Thanks.

Answer 1

One property is that $n\not\equiv2\pmod3$.

This is because $n\equiv\color\red2\pmod3\implies4n+1\equiv4\cdot\color\red2+1\equiv9\equiv0\pmod3$.

Answer 2

This is just a slight generalisation of barak manos' answer.

Let $p_n$ be the $n$'th prime equal to $3\pmod 4$. We now know that: $$n\not\equiv \frac{p_n^2-1}{4}\pmod {p_n}$$ Because otherwise: $$4n+1\equiv p_n^2\equiv 0\pmod {p_n}$$