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Consider the twin primes. So we consider $3,5,7,11,13,17,19,29,31,...$

Call the $n$-th element in the sequence $t(n)$.

Now consider the Sophie Germain primes ( if $p$ is prime, then so is $2p + 1$. ONLY count the $p$ not the $2p+1$ )

Call the $n$-th element of that sequence $s(n)$.

Now for $1000 < n$ a friend noticed a strange pattern.

$$ \frac{\ln(n)}{\ln(2n+1)} 5 < \frac{2 s(n)}{t(n)} < \frac{\ln(2n+1)}{\ln(n)} 5 $$

You can probably guess where the $2n+1$ more or less comes from somehow.

Now for the analogue case; the sexy primes we can easily see why there are probably $2$ times more sexy primes than prime twins. ( mod argument )

So I wonder what the argument is is for the mysterious $5$ ?

There is no proof for the infinitude of twin primes or sophie germain primes, let alone their asymptotics but I ask for the argument of that $5$ not a proof.

I assume there must be reason.

This implies ofcourse that the limit of $2 s(n)/t(n) = 5$ and for instance we see that $s(4800)/t(4800) = 4.966 $.

mick
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    First of all, we should try to extend the range where this double inequality holds. $n=1000$ is very small. – Peter Aug 14 '20 at 11:49
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    I get $s(4800)=565289$ , $t(4800)=529577$ which gives another ratio. How does the sequence $s(n)$ look like ? Or did I miss something else ? – Peter Aug 14 '20 at 12:02
  • @Peter Also I tested many $n$ ofcourse not just a few small above $1000$ – mick Aug 14 '20 at 20:44
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    I think Peter's numbers are correct. The graphs at https://oeis.org/A005384/graph and https://oeis.org/A001359/graph do not appear to differ by a multiple of 5. In any case, the Bateman-Horn conjecture predicts the exact same asymptotic for these two sequences. – Ravi Fernando Aug 14 '20 at 20:51
  • Ok I modified things. All should be clear now. – mick Aug 14 '20 at 21:25
  • $\frac{\ln(2n+1)}{\ln(n)} 5 = 5+\frac{\ln(32)}{\ln(n)}+...$ – Roddy MacPhee Apr 09 '21 at 01:24
  • @RoddyMacPhee yes. But why do you say that ? – mick Apr 09 '21 at 11:16
  • People might also like this :

    https://math.stackexchange.com/questions/3179484/a-question-on-the-density-of-sophie-germain-primes

     – mick Jun 14 '23 at 22:15

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