bi-twin chain

In number theory, a bi-twin chain of length k + 1 is a sequence of natural numbers

: $n-1,n+1,2n-1,2n+1,\; \backslash dots,\; 2^k\; n\; -\; 1,\; 2^k\; n\; +\; 1\; \backslash ,$

in which every number is prime.Eric W. Weisstein, CRC Concise Encyclopedia of Mathematics, CRC Press, 2010, page 249.

The special case, when the four numbers $n-1,n+1,2n-1,2n+1$ are all primes, they are called bi-twin primes,[http://www.primenumbers.net/Henri/fr-us/BiTwinRec.htm BiTwin records] such n values are

:6, 30, 660, 810, 2130, 2550, 3330, 3390, 5850, 6270, 10530, 33180, 41610, 44130, 53550, 55440, 57330, 63840, 65100, 70380, 70980, 72270, 74100, 74760, 78780, 80670, 81930, 87540, 93240, … {{OEIS|A066388}}

Except 6, all of these numbers are divisible by 30.

The numbers $n-1,\; 2n-1,\; \backslash dots,\; 2^kn\; -\; 1$ form a Cunningham chain of the first kind of length $k\; +\; 1$, while $n+1,\; 2n\; +\; 1,\; \backslash dots,\; 2^kn\; +\; 1$ forms a Cunningham chain of the second kind. Each of the pairs $2^in\; -\; 1,\; 2^in+\; 1$ is a pair of twin primes. Each of the primes $2^in\; -\; 1$ for $0\; \backslash le\; i\; \backslash le\; k\; -\; 1$ is a Sophie Germain prime and each of the primes $2^in\; -\; 1$ for $1\; \backslash le\; i\; \backslash le\; k$ is a safe prime.