Is there something special about primes p where 2p-1 is also prime? They aren't rare apparently: 2 and 3, 3 and 5, 7 and 13, 19 and 37 or 307 and 613.

q = 2p-1 fulfills one half of the "strong prime" requirement in cryptography, i.e. q+1 has only large prime factors, same as for SG q-1 does.

It could be applied to Proth primes:

k * 2^n + 1 = p
k * 2^(n+1) + 1 = 2p - 1

So why aren't they investigated the same? Once more I guess I'm missing something, but what? :)
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1281979 * 2^485014 + 1 is prime ... no further hits up to: n = 5,700,000

Sophie Germain primes are related to the first case of Fermat's last theorem, but Cunningham chains of the second kind have no known mathematical property.
See https://primes.utm.edu/top20/page.php?id=20

Ok, thanks.
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1281979 * 2^485014 + 1 is prime ... no further hits up to: n = 5,700,000