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I think what you ask about is a partial case of Bateman–Horn conjecture for polynomials $f_1(x)=x,f_2(x)=4x+1$. The expected value of $C$ equals $(1-1/2)\prod_{p>2}\left\{(1-2/p)(1-1/p)^{-2}\right\}$, product is taken over primes. Roughly speaking, $1-1/2$ is the probability that $n$ is even and $1-2/p$ for odd prime $p$ is the probability that $n$ and $4n+1$ both are not divisible by $p$.

I think what you ask about is a partial case of Bateman–Horn conjecture for polynomials $f_1(x)=x,f_2(x)=4x+1$. The expected value of $C$ equals $$(1-1/2)(1-1/2)^{-2}\prod_{p>2}\left\{(1-2/p)(1-1/p)^{-2}\right\},$$ product is taken over primes. Roughly speaking, $1-1/2$ is the probability that $n$ is even and $1-2/p$ for odd prime $p$ is the probability that $n$ and $4n+1$ both are not divisible by $p$.

I think what you ask about is a partial case of Bateman–Horn conjecture for polynomials $f_1(x)=x,f_2(x)=4x+1$. The expected value of $C$ equals $(1-1/2)\prod_{p>2}(1-2/p)$, product is taken over primes. Roughly speaking, $1-1/2$ is the probability that $n$ is even and $1-2/p$ for odd prime $p$ is the probability that $n$ and $4n+1$ both are not divisible by $p$.

I think what you ask about is a partial case of Bateman–Horn conjecture for polynomials $f_1(x)=x,f_2(x)=4x+1$. The expected value of $C$ equals $(1-1/2)\prod_{p>2}\left\{(1-2/p)(1-1/p)^{-2}\right\}$, product is taken over primes. Roughly speaking, $1-1/2$ is the probability that $n$ is even and $1-2/p$ for odd prime $p$ is the probability that $n$ and $4n+1$ both are not divisible by $p$.

I think what you ask about is a partial case of https://en.wikipedia.org/wiki/Bateman%E2%80%93Horn_conjecture

I think what you ask about is a partial case of Bateman–Horn conjecture for polynomials $f_1(x)=x,f_2(x)=4x+1$. The expected value of $C$ equals $(1-1/2)\prod_{p>2}(1-2/p)$, product is taken over primes. Roughly speaking, $1-1/2$ is the probability that $n$ is even and $1-2/p$ for odd prime $p$ is the probability that $n$ and $4n+1$ both are not divisible by $p$.