cp's OEIS Frontend

In principle, moderate values should appear infinitely many times, by analogy with twin primes hypothesis. For example, a(n) = 44 for n = 9, 17, 206, 1604467, 12905293, 18008874, 26545460, 32655424, 57848470, 58313630, 59022635, 66275281, 81581956, 123780499, 160884754, 167797255, 179786560, 181569324, 239542290, ... - Zak Seidov, Sep 14 2014, edited by M. F. Hasler, Dec 03 2018

According to the k-tuple conjecture, any admissible k-tuple of primes occurs with calculable nonzero asymptotic density, i.e., in particular, infinitely many times. For k = 11, number of primes in the interval [prime(n), prime(n+10)], the smallest possible diameter of a k-tuple is A008407(11) = 36, and there are A083409(11) = 2 such constellations: {0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 36}, first occurring at A213646(1) = 1418575498573, and {0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36}, first occurring at A213647(1) = 11. The combined list { prime(n) | a(n) = 36 } is A257129. - M. F. Hasler, Dec 03 2018

Prime quadruplets (p, p+2, p+6, p+8) are densest permissible constellations of 4 primes (A007530). By the Hardy-Littlewood k-tuple conjecture, average gaps between prime k-tuples are O(log^k(p)), with k=4 for quadruplets. If a gap is larger than all preceding gaps, we call it a maximal gap, or a record gap. Maximal gaps may be significantly larger than average gaps. This sequence suggests that maximal gaps between prime quadruplets are O(log^5(p)). - Alexei Kourbatov, Jan 04 2012

Prime sextuplets (p, p+4, p+6, p+10, p+12, p+16) are densest permissible constellations of 6 primes. Average gaps between sextuplets (and, more generally, between prime k-tuples) can be deduced from the Hardy-Littlewood k-tuple conjecture and are O(log^6(p)). Maximal gaps may be significantly larger than average gaps; this sequence suggests that maximal gaps are O(log^7(p)).

A200504 lists initial primes in sextuplets preceding the maximal gaps. A233426 lists the corresponding primes at the end of the maximal gaps.

The (wrong) old name was: Largest number of pairwise coprime numbers that can occur in an interval of length n. - Wolfdieter Lang, Oct 10 2017

Conjecturally, a(n) is the largest number of primes that occurs on an infinite number of intervals of n consecutive integers. The conjecture is apparently due to Dickson; Hardy & Littlewood's Conjecture B concerns only pairs (p, p + 2n).

According to the link at www.opertech.com, a(3159) >= 447 > 446 = pi(3159). The k-tuples conjecture then implies that, for an infinitude of n, the interval [n+1, n+3159] includes 447 primes. For these n, pi(n+3159) >= pi(n)+447 > pi(n)+446 = pi(n)+pi(3159), contradicting the conjecture that pi(x+y) <= pi(x)+pi(y). - David W. Wilson, May 23 2005

From Wolfdieter Lang, Oct 10 2017: (Start)

The following admissibility definition is adapted from the Hensley and Richards [H-R] or Richards [R] links. A k-tuple B_k = [b_1, b_2, ..., b_k] of integers with 0 <= b_1 < b_2 < ... < b_k is admissible if, for each prime p, there exists at least one congruence class modulo p which contains none of the B_k members. Because complete residue systems modulo p are equivalent under translation one can consider the length n interval [0, 1, ..., n-1] and admissible k-tuples starting with 0. The prime p = 2 allows then only even tuple numbers from I_n = [0, 2, ..., floor((n-1)/2)]. Only primes p <= k have to be tested.

a(n) is then the maximal k for which there is at least one such admissible B_k tuple from the interval I_n. This function a(n) is called rho^* in (H-R) and (R). It has been given as rhobar in the Schinzel - Sierpiński link, Théorème 1, p. 201.

Note that there are also admissible k-tuples from members of [0, 1, ..., n-1] which do not start with 0. Such tuples are translations of the ones starting with 0. E.g., [1, 3] is an admissible 2-tuple for any [0, 1,..., n-1] interval with n >= 3, but it is a translation of the considered [0, 2] tuple.

For the multiplicities of k see A047947(k), for k >= 1.

For the smallest k such that a(k) = n see A020497(n), for k >= 1.

For the number of all admissible k-tuples from the interval I_n starting with 0 see the array A292224(n, k), with k = 1..a(n), which has been given in the Engelsma link, Table 2, p. 27.

One of the Hardy-Littlewood conjectures (the prime tuple conjecture, see also conjecture (B) given by [H-R] and [R], and Ribenboim, hypothesis (D_1), p. 373, from the Dickson conjecture) is that there are infinitely many primes with gaps defined by any admissible B_k tuple, that is, all p, p + b_2, ..., p + b_k are prime for infinitely many primes p, for k >= 2. For k = 1 this is well known.

(End)

For a proper definition see the cross-references.

For the definition of n-tuplet and minimal differences without the symmetry restriction, see A008407. In particular, a(n) >= A008407(n).

An n-tuplet (p(1),...,p(n)) is symmetric if p(k) + p(n+1-k) is the same for all k=1,2,...,n (cf. A175309).

Smallest primes starting a shortest symmetric n-tuplet are given in A266512.

For odd n, a(n) is divisible by 12.

a(n) >= A210439(n). Equals A210439(n) at n=2,4,6, i.e., at those n for which there is only one type of prime n-tuplets (admissible prime n-tuples of minimal span). The corresponding minimal span (diameter) is given by A008407(n).

See A210439 for more information, references and links.

From Hugo Pfoertner, Oct 21 2021: (Start)

There are two options for choosing a(8):

Either one interprets "latest occurrence" as the largest number of 8-tuplets before the Hardy-Littlewood (H-L) prediction is exceeded, or one selects the larger value of the first 8-tuplet term causing the first crossing.

In the first case, 40634356 8-tuplets of the type p + [0, 2, 6, 12, 14, 20, 24, 26] are required before the H-L prediction is exceeded with an 8-tuplet 523250002674163757 + [0, 2, 6, ...].

In the second case, 20316822 8-tuplets of type p + [0, 6, 8, 14, 18, 20, 24, 26] are needed to reach the first crossing of the H-L prediction. The corresponding 8-tuplet has 750247439134737983 as first term.

The interchanging is a consequence of the different H-L constants for the two tuplet types, 475.36521.. vs. 178.261954.., which have a ratio of 8/3 to one another.

Since the H-L constant for the "earliest occurrence" A210439(8) is 178.26.., this speaks in favor of a choice from the two possibilities, which uses the same H-L constant, i.e., the occurrence with the larger tuplet start and not the occurrence with the larger number of required tuplets, for which a separate sequence A348053 is created. (End)

Given a(i) for 1 <= i < n, a(n) is the smallest number > a(n-1) such that, for every prime p, the set {a(i) mod p : 1<=i<=n} has at most p-1 elements. Assuming Schinzel's hypothesis H, an equivalent statement is that a(n) is minimal such that there are infinitely many primes p with p+a(i) prime for 1 <= i <= n.

For every n, a(n) is not congruent to 1 (mod 2), nor to 1 (mod 3), nor to 4 (mod 5), nor to 3 (mod 7), ...

Note that this sequence does not always give the minimal difference between the first and last of n consecutive large primes, A008407. E.g., a(6)=18 but the 6 consecutive primes 97, 101, 103, 107, 109, 113 give the minimal difference of 16.

Conjecturally, a(n) is the smallest number such that n primes occur infinitely often among (x+a(1), ...,x+a(n)).

From M. F. Hasler, Nov 25 2024: (Start)

For a given prime p, if r is the only residue (mod p) not among {a(1), ..., a(n)} (mod p) for some index n, then no term of the sequence can be congruent to r (mod p).

(Instead of a(1...n), one can consider any collection of terms.) - Examples:

(1) p = 2, r = 0, n = 1: No term can be congruent to 0 (mod 2), i.e., even.

(2) p = 3, r = 2, n = 2: No term may be congruent to 2 (mod 3).

(3) p = 5, r = 0, n = 4: No term may be a multiple of 5.

(4) p = 7, r = 4, n = 6: No term may be congruent to 4 (mod 7).

(5) p = 11, r = 6, n = 11: No term may be congruent to 6 (mod 11). (End)

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.