Prime Constellations

n-uples of consecutive prime numbers consecutive whose differences are given.

Example

(487, 491, 499, 503) and the differences 4 8 4
(487, 491, 499, 503) is a 4-uple of consecutive primes and the differences are, in order,
491-487 = 4, 499-491 = 8 and 503-499 = 4.
Use star * as joker, to represent any nonnull value.
The search is carried out in the table of the prime numbers from 1 to 500 000

Definitions

In the set of positive integers:
n is prime exactly two dividers (1 and n).
Twin primes have 2 for difference, prime cousins have 4 for difference.

Conjectures

One conjectures that there is an infinity of pairs of prime numbers twins.
The conjecture of Polignac is that for all k>0 fixed, there exists an infinity of pairs of
prime numbers of the form {p, p+2k}.
The conjecture of the constellations is that for any possible form given, there is an infinity
of constellations.
(p, p+2, p+4) is not a possible form, except for (3, 5, 7).
A form is possible when all the numbers of the sequence do not occupy all the classes
of congruence of a prime number.

Incompatibles conjectures:

The conjecture of the constellations and the conjecture of decrease of the densities
are incompatible
(D. Hensley and Ian Richard 1973)

Constellations

(Differences between consecutive primes)
Pattern for a constellation : 



Differences for a progression : 
(The primes are not necessarely consecutives)

While taking as model '2', one obtains the 4565 pairs of twins lower than 500 000.
The pairs of prime cousins are obtained by using the models '4' and '2 2'.
Which are the first sexy ones?
The corresponding sequences of the encyclopaedia of Sloane are A001359, A023200 et A023201.
See also the sequences A029710, A031924 ou A031926