Bi-twin chain

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

${\displaystyle n-1,n+1,2n-1,2n+1,\dots ,2^{k}n-1,2^{k}n+1\,}$

in which every number is prime.^[1]

The special case, when the four numbers ${\displaystyle n-1,n+1,2n-1,2n+1}$ are all primes, they are called bi-twin primes,^[2] 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, … (sequence A066388 in the OEIS)

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

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