This site is supported by donations to The OEIS Foundation.
Prime constellations
From OeisWiki
Here, we make a distinction between a prime constellation and a prime cluster.^{[1]}
Contents 
Prime clusters
A prime cluster, also called a prime ktuple (double, triple, quadruple, ...) is a strictly increasing sequence of primes such that the difference between the first and last is not necessarily minimal, i.e. some primes could be skipped over. More precisely, a prime ktuple is a strictly increasing sequence of primes with , where is not necessarily the smallest number for which there exist integers and for every prime , not all the residues modulo are represented by .^{[1]}
Nonadmissible prime clusters
A nonadmissible prime cluster, also called a nonadmissible prime ktuple, is a prime cluster such that for some prime , all the residues modulo are represented by . Only a finite number of nonadmissible prime clusters may appear at the beginning of a prime cluster sequence.
For each , this definition excludes a finite number of clusters at the beginning of the prime number sequence. For example, (97, 101, 103, 107, 109) satisfies the conditions of the definition of a prime 5tuplet, but (3, 5, 7, 11, 13) does not because all three residues modulo 3 are represented.^{[1]}
Prime constellations
A prime constellation, also called a prime ktuplet (doublet, triplet, quadruplet, ...) is a maximally dense prime cluster, i.e. a sequence of consecutive primes, i.e. such that the difference between the first and last is minimal, i.e. no primes could be skipped over. More precisely, a prime tuplet is a sequence of consecutive primes, i.e. with , where is the smallest number for which there exist integers and, for every prime , not all the residues modulo are represented by .^{[1]}
Prime pairs
Twin primes
Twin primes: prime pairs 

Cousin primes
Cousin primes: prime pairs 

Prime triplets
Prime triplets: are all prime 

Prime triplets: are all prime 

Prime triplets: or are all prime 

Prime quadruplets
Prime quadruplets: are all prime. 

Prime 5 tuplets
Prime 5 tuplets: are all prime 

Prime 5 tuplets: are all prime 

Notes
 ↑ ^{1.0} ^{1.1} ^{1.2} ^{1.3} Tony Forbes, Prime Clusters and Cunningham Chains, Mathematics of Computation, Volume 68, Number 228, pp. 17391747.
 ↑ Weisstein, Eric W., Twin Prime Conjecture, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/TwinPrimeConjecture.html]
 ↑ ^{3.0} ^{3.1} Weisstein, Eric W., kTuple Conjecture, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/kTupleConjecture.html]