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According to the conjecture in A236460, this sequence should have infinitely many terms.

See A236464 for a similar sequence.

The first pattern (0,2) is for twin primes (p,p+2). Row n contains A083409(n) patterns, each one consisting of 0 followed by n-1 terms. In each row the patterns are in lexicographic order.

These numbers (in a slightly different order) appear in Table 1 of the paper by Tony Forbes. Sequence A186702 gives the least prime starting a given pattern.

Conjecture: a(n) > 0 for all n > 211.

This implies that there are infinitely many primes p with {prime(p), prime(p) + 4, prime(p) + 6} a prime triple. See A236462 for such primes p.

Primes p such that the second differences of p and the next 2 primes is never positive.

Superset of A022005.

Does not intersect A022004.

Prime triples (p, p+4, p+6) are one of the two types of densest permissible constellations of 3 primes. Maximal gaps between triples of this type are listed in A201596; see more comments there.

The row lengths are given by A023193 (the rhobar function of Schinzel and Sierpiński called rho* by Hensley and Richards).

This irregular triangle has already been considered by Engelsma, see Table 2, for n=1..56, p. 27.

A k-tuple of integers B_k = [b_1, ..., b_k] with 0 = b_1 < b_2 < ... < b_k <= n-1 is called admissible if for each prime p there exists at least one congruence class modulo p which contains none of the B_k elements. (This corresponds to the alternative definition of Hensley and Richards, p. 378 (*) or Richards, p. 423, 1.5 Definition and (*).) Note that the definition of "admissibility" is translation invariant (see the Note by Richards, p. 424, which is obvious from the translation equivalence of complete residue systems modulo p). Therefore the interval I_n = [0, n-1] of length n has been chosen. The b_1 = 0 choice is conventional. Without this choice other admissible k-tuples are obtained by translation as long as b_k + a < n-1. E.g., for n = 8 and k = 3 the tuple [1, 5, 7] is admissible and a translation of the considered tuple [0, 4, 6].

Only primes p <= k have to be tested to decide on the admissibility of a B_k tuple because for larger k there is always some residue class which contains none of the k members of B_k.

Because p = 2 already forbids even and odd numbers to appear in B_k for k >= 2, one can for the admissibility test eliminate all odd numbers in the chosen I_n. Therefore, only Ieven_n:= [0, 2, ..., 2*floor((n-1)/2)] =: 2*[0, 1, ..., floor((n-1)/2)] need be considered. B_1 = [0] is admissible for all n >= 1.

Because only the interval Ieven_n is of relevance, there will occur repetitions for admissible tuples for n if n = 2*k+1 and n = 2*k+2.

With the set B_k(p) = B_k (mod p) := {0, b_1 (mod p), ..., b_k (mod p)} the criterion for admissibility can be written as p - #(B_k(p)) > 0, for all primes 3 <= p <= k (because there are p congruence classes defined by smallest nonnegative complete residue system [0, 1, ..., p-1]).

Admissible tuples (starting with 0) with least b_k - b_1 = b_k value give rise to prime k-constellations of diameter b_k. E.g., for k = 2 the admissible tuple [0, 4] does not lead to a prime 2-constellation for n >= 5; [0, 6] is out for n >= 7, ... . But there are two prime 3-constellations given by [0, 2, 6] and [0, 4, 6] for n >= 7.

Row sums are in A292225, that is, total number of admissible tuples starting with 0 from the interval I_n = [0, n-1].

Prime triples (p, p+4, p+6) are one of the two types of densest permissible constellations of 3 primes. Maximal gaps between triples of this type are listed in A201596; see more comments there.

A prime triple is a set of three prime numbers of the form (p, p+2, p+6) or (p, p+4, p+6).

Initial members p of prime triples of the form (p, p+4, p+6) are congruent to 7 or 13 (mod 30).

Also called prime triples of the second kind.

Equivalently, least k > 0 such that either 10^n + k + {0, 2, 6} or 10^n + k + {0, 4, 6} are primes.

The initial term, index n = 0, is the only even term and the only case where the last member of the triplet has one digit more than the first member. The value a(0) = 4 correspond to the prime triplet (5, 7, 11). We do not consider the triplets (2, 3, 5) or (3, 5, 7) which come earlier but do not follow the standard pattern.

With a prime triple (p,p+4,p+6), the number a(n) = 2*p*(p+6) is always in the sequence, f( f( 2*p*(p+6) )) = f( 2*(p+4) ) = p+6. Such prime triples can be found in sequence A022005.

As long as two successive triples (p1,p1 + 4,p1 + 6) and (p2,p2 + 4,p2 + 6) of A022005 have p2 < 1.2*p1, no other numbers occur in the sequence between a(n1) and a(n2), this holds at least for larger p1 > 500. Other types of prime sets occurring in the sequence: (p,p+4,3p-4) with F( F( (p+4)*(3p-4))) = F( 4p ) = p + 4 (p,p+6,p+8) with F( F( 4*p*(p+8) )) = F( 2*(p+6) ) = p + 8.

Large examples of (p,p+4,++6)-triples: (108748629354*4436*3251#*(4436*3251#+1)+210)*(4436*3251#-1)/35 + 7, + 11, + 13 (4135 digits, David Broadhurst) (18599651274*4436*3251#*(4436*3251#+1)+210)*(4436*3251#-1)/35 + 7, + 11, +13 (4134 digits, David Broadhurst) Record examples of prime triples can be found on Tony Forbes's web site. There are triples of type (p,p+4,p+6) too.