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 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.