A022006 -id:A022006 - cp's OEIS frontend

A257374 Numbers n such that n, n+4, n+10, n+12, n+16, n+22, n+24, n+30, n+36, n+40, n+42, n+46, n+52, n+54, n+60, n+64 and n+66 are all prime.

734975534793324512717947, 753314125249587933791677, 1341829940444122313597407

Offset: 1

Tim Johannes Ohrtmann, Apr 21 2015

Tony Forbes and Norman Luhn, k-tuplets

Initial members of all of the first prime k-tuplets:
twin primes: A001359.
prime triples: A007529 out of A022004, A022005.
prime quadruplets: A007530.
prime 5-tuples: A086140 out of A022007, A022006.
prime sextuplets: A022008.
prime septuplets: A257124 out of A022009, A022010.
prime octuplets: A065706 out of A022011, A022012, A022013.
prime nonuplets: A257125 out of A022547, A022548, A022545, A022546.
prime decaplets: A257127 out of A027569, A027570.
prime 11-tuplets: A257129 out of A213646, A213647.
prime 12-tuplets: A257131 out of A213601, A213645.
prime 13-tuplets: A257135 out of A214947, A257137, A257138, A257139, A257140, A257141.
prime 14-tuplets: A257166 out of A257167, A257168.
prime 15-tuplets: A257169 out of A257304, A257305, A257306, A257307.
prime 16-tuplets: A257308 out of A257369, A257370.
prime 17-tuplets: A257373 out of this sequence, A257375, A257376, A257377.

a(3) from Norman Luhn, Oct 27 2021

A257375 Numbers n such that n, n+4, n+6, n+10, n+16, n+18, n+24, n+28, n+30, n+34, n+40, n+46, n+48, n+54, n+58, n+60 and n+66 are all prime.

Original entry on oeis.org

13, 47624415490498763963983, 78314167738064529047713, 83405687980406998933663, 110885131130067570042703, 163027495131423420474913

Offset: 1

Tim Johannes Ohrtmann, Apr 21 2015

Tony Forbes k-tuplets

Initial members of all of the first prime k-tuplets:
twin primes: A001359.
prime triples: A007529 out of A022004, A022005.
prime quadruplets: A007530.
prime 5-tuples: A086140 out of A022007, A022006.
prime sextuplets: A022008.
prime septuplets: A257124 out of A022009, A022010.
prime octuplets: A065706 out of A022011, A022012, A022013.
prime nonuplets: A257125 out of A022547, A022548, A022545, A022546.
prime decaplets: A257127 out of A027569, A027570.
prime 11-tuplets: A257129 out of A213646, A213647.
prime 12-tuplets: A257131 out of A213601, A213645.
prime 13-tuplets: A257135 out of A214947, A257137, A257138, A257139, A257140, A257141.
prime 14-tuplets: A257166 out of A257167, A257168.
prime 15-tuplets: A257169 out of A257304, A257305, A257306, A257307.
prime 16-tuplets: A257308 out of A257369, A257370.
prime 17-tuplets: A257373 out of A257374, this sequence, A257376, A257377.

A257376 Numbers n such that n, n+6, n+8, n+12, n+18, n+20, n+26, n+32, n+36, n+38, n+42, n+48, n+50, n+56, n+60, n+62 and n+66 are all prime.

Original entry on oeis.org

1620784518619319025971, 2639154464612254121531, 3259125690557440336631, 124211857692162527019731

Offset: 1

Tim Johannes Ohrtmann, Apr 21 2015

Tony Forbes k-tuplets

Initial members of all of the first prime k-tuplets:
twin primes: A001359.
prime triples: A007529 out of A022004, A022005.
prime quadruplets: A007530.
prime 5-tuples: A086140 out of A022007, A022006.
prime sextuplets: A022008.
prime septuplets: A257124 out of A022009, A022010.
prime octuplets: A065706 out of A022011, A022012, A022013.
prime nonuplets: A257125 out of A022547, A022548, A022545, A022546.
prime decaplets: A257127 out of A027569, A027570.
prime 11-tuplets: A257129 out of A213646, A213647.
prime 12-tuplets: A257131 out of A213601, A213645.
prime 13-tuplets: A257135 out of A214947, A257137, A257138, A257139, A257140, A257141.
prime 14-tuplets: A257166 out of A257167, A257168.
prime 15-tuplets: A257169 out of A257304, A257305, A257306, A257307.
prime 16-tuplets: A257308 out of A257369, A257370.
prime 17-tuplets: A257373 out of A257374, A257375, this sequence, A257377.

a(1) corrected by Tim Johannes Ohrtmann, Dec 17 2015

A257377 Numbers n such that n, n+2, n+6, n+12, n+14, n+20, n+24, n+26, n+30, n+36, n+42, n+44, n+50, n+54, n+56, n+62 and n+66 are all prime.

Original entry on oeis.org

17, 37630850994954402655487, 53947453971035573715707, 174856263959258260646207, 176964638100452596444067, 207068890313310815346497, 247620555224812786876877, 322237784423505559739147

Offset: 1

Tim Johannes Ohrtmann, Apr 21 2015

Tony Forbes k-tuplets

Initial members of all of the first prime k-tuplets:
twin primes: A001359.
prime triples: A007529 out of A022004, A022005.
prime quadruplets: A007530.
prime 5-tuples: A086140 out of A022007, A022006.
prime sextuplets: A022008.
prime septuplets: A257124 out of A022009, A022010.
prime octuplets: A065706 out of A022011, A022012, A022013.
prime nonuplets: A257125 out of A022547, A022548, A022545, A022546.
prime decaplets: A257127 out of A027569, A027570.
prime 11-tuplets: A257129 out of A213646, A213647.
prime 12-tuplets: A257131 out of A213601, A213645.
prime 13-tuplets: A257135 out of A214947, A257137, A257138, A257139, A257140, A257141.
prime 14-tuplets: A257166 out of A257167, A257168.
prime 15-tuplets: A257169 out of A257304, A257305, A257306, A257307.
prime 16-tuplets: A257308 out of A257369, A257370.
prime 17-tuplets: A257373 out of A257374, A257375, A257376, this sequence.

A201062 Record (maximal) gaps between prime 5-tuples (p, p+4, p+6, p+10, p+12).

Original entry on oeis.org

90, 1770, 2190, 10080, 24360, 35910, 156750, 208620, 304920, 306390, 328020, 422190, 526350, 639330, 706860, 866460, 1030770, 1111620, 1147440, 1151100, 1447530, 1769670, 1793070, 2024610, 2320170, 2335080, 2403570

Offset: 1

Alexei Kourbatov, Nov 26 2011

nonn

Prime quintuplets (p, p+4, p+6, p+10, p+12) are one of the two types of densest permissible constellations of 5 primes (A022006 and A022007). Average gaps between prime k-tuples can be deduced from the Hardy-Littlewood k-tuple conjecture and are O(log^k(p)), with k=5 for quintuplets. 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 quintuplets are O(log^6(p)).

A201063 lists initial primes in quintuplets (p, p+4, p+6, p+10, p+12) preceding the maximal gaps. A233433 lists the corresponding primes at the end of the maximal gaps.

			The gap of 90 between quintuplets starting at p=7 and p=97 is the very first gap, so a(1)=90. The gap of 1770 between quintuplets starting at p=97 and p=1867 is a maximal gap - larger than any preceding gap; therefore a(2)=1770. The gap after p=1867 is smaller, so a new term is not added.

Alexei Kourbatov, Table of n, a(n) for n = 1..71
Tony Forbes, Prime k-tuplets
G. H. Hardy and J. E. Littlewood, Some problems of 'Partitio numerorum'; III: on the expression of a number as a sum of primes, Acta Mathematica, Vol. 44, pp. 1-70, 1923.
Alexei Kourbatov, Maximal gaps between prime 5-tuples (graphs/data up to 10^15)
A. Kourbatov, Maximal gaps between prime k-tuples: a statistical approach, arXiv preprint arXiv:1301.2242 [math.NT], 2013 and J. Int. Seq. 16 (2013) #13.5.2
Alexei Kourbatov, Tables of record gaps between prime constellations, arXiv preprint arXiv:1309.4053 [math.NT], 2013.
Alexei Kourbatov, The distribution of maximal prime gaps in Cramer's probabilistic model of primes, arXiv preprint arXiv:1401.6959 [math.NT], 2014.
Alexei Kourbatov and Marek Wolf, Predicting maximal gaps in sets of primes, arXiv preprint arXiv:1901.03785 [math.NT], 2019.
Eric W. Weisstein, k-Tuple Conjecture

Cf. A022007 (prime 5-tuples p, p+4, p+6, p+10, p+12), A113274, A113404, A200503, A201596, A201598, A201073, A201051, A201251, A202281, A202361, A201063, A002386, A233433.

DeleteDuplicates[Differences[Select[Partition[Prime[Range[10^7]],5,1],Differences[#]=={4,2,4,2}&][[;;,1]]],GreaterEqual] (* The program generates the first 18 terms of the sequence. *) (* Harvey P. Dale, Apr 20 2025 *)

(1) Upper bound: gaps between prime 5-tuples are smaller than 0.0987*(log p)^6, where p is the prime at the end of the gap.
(2) Estimate for the actual size of the maximal gap that ends at p: maximal gap ~ a(log(p/a)-0.4), where a = 0.0987*(log p)^5 is the average gap between quintuplets near p, as predicted by the Hardy-Littlewood k-tuple conjecture.
Formulas (1) and (2) are asymptotically equal as p tends to infinity. However, (1) yields values greater than all known gaps, while (2) yields "good guesses" that may be either above or below the actual size of known maximal gaps.
Both formulas (1) and (2) are derived from the Hardy-Littlewood k-tuple conjecture via probability-based heuristics relating the expected maximal gap size to the average gap. Neither of the formulas has a rigorous proof (the k-tuple conjecture itself has no formal proof either). In both formulas, the constant ~0.0987 is reciprocal to the Hardy-Littlewood 5-tuple constant 10.1317...

A201073 Record (maximal) gaps between prime 5-tuples (p, p+2, p+6, p+8, p+12).

Original entry on oeis.org

6, 90, 1380, 14580, 21510, 88830, 97020, 107100, 112140, 301890, 401820, 577710, 689850, 846210, 857010, 986160, 1655130, 2035740, 2266320, 2467290, 2614710, 3305310, 3530220, 3880050, 3885420, 5290440, 5713800, 6049890

Offset: 1

Alexei Kourbatov, Nov 26 2011

nonn

Prime quintuplets (p, p+2, p+6, p+8, p+12) are one of the two types of densest permissible constellations of 5 primes (A022006 and A022007). Average gaps between prime k-tuples can be deduced from the Hardy-Littlewood k-tuple conjecture and are O(log^k(p)), with k=5 for quintuplets. If a gap is larger than any preceding gap, 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 are O(log^6(p)).

A201074 lists initial primes in quintuplets (p, p+2, p+6, p+8, p+12) preceding the maximal gaps. A233432 lists the corresponding primes at the end of the maximal gaps.

			The initial four gaps of 6, 90, 1380, 14580 (between quintuplets starting at p=5, 11, 101, 1481, 16061) form an increasing sequence of records. Therefore a(1)=6, a(2)=90, a(3)=1380, and a(4)=14580. The next gap (after 16061) is smaller, so a new term is not added.

Alexei Kourbatov, Table of n, a(n) for n = 1..64
Tony Forbes, Prime k-tuplets
G. H. Hardy and J. E. Littlewood, Some problems of 'Partitio numerorum'; III: on the expression of a number as a sum of primes, Acta Mathematica, Vol. 44, pp. 1-70, 1923.
Alexei Kourbatov, Maximal gaps between prime 5-tuples (graphs/data up to 10^15)
A. Kourbatov, Maximal gaps between prime k-tuples: a statistical approach, arXiv preprint arXiv:1301.2242, 2013 and J. Int. Seq. 16 (2013) #13.5.2
Alexei Kourbatov, Tables of record gaps between prime constellations, arXiv preprint arXiv:1309.4053 [math.NT], 2013.
Alexei Kourbatov, The distribution of maximal prime gaps in Cramer's probabilistic model of primes, arXiv preprint arXiv:1401.6959 [math.NT], 2014.
Alexei Kourbatov and Marek Wolf, Predicting maximal gaps in sets of primes, arXiv preprint arXiv:1901.03785 [math.NT], 2019.
Eric W. Weisstein, k-Tuple Conjecture

Cf. A022006 (prime 5-tuples p, p+2, p+6, p+8, p+12), A113274, A113404, A200503, A201596, A201598, A201051, A201251, A202281, A202361, A201062, A201074, A002386, A233432.

(1) Upper bound: gaps between prime 5-tuples are smaller than 0.0987*(log p)^6, where p is the prime at the end of the gap.
(2) Estimate for the actual size of the maximal gap that ends at p: maximal gap ~ a(log(p/a)-0.4), where a = 0.0987*(log p)^5 is the average gap between quintuplets near p, as predicted by the Hardy-Littlewood k-tuple conjecture.
Formulas (1) and (2) are asymptotically equal as p tends to infinity. However, (1) yields values greater than all known gaps, while (2) yields "good guesses" that may be either above or below the actual size of known maximal gaps.
Both formulas (1) and (2) are derived from the Hardy-Littlewood k-tuple conjecture via probability-based heuristics relating the expected maximal gap size to the average gap. Neither of the formulas has a rigorous proof (the k-tuple conjecture itself has no formal proof either). In both formulas, the constant ~0.0987 is reciprocal to the Hardy-Littlewood 5-tuple constant 10.1317...

A078969 Primes p such that the differences between the 5 consecutive primes starting with p are (6,6,6,4).

Original entry on oeis.org

3301, 15901, 18211, 30091, 53611, 71341, 77551, 80911, 89101, 120811, 252151, 285451, 292471, 294781, 344251, 601801, 616501, 744811, 792691, 809821, 908521, 912391, 1152631, 1154221, 1279801, 1376491, 1398031, 1455361, 1464271, 1500511, 1503031, 1555111, 1594261

Offset: 1

Labos Elemer, Dec 19 2002

nonn

Equivalently, primes p such that p, p+6, p+12, p+18 and p+22 are consecutive primes.

			30091 is in the sequence since 30091, 30097 = 30091 + 6, 30103 = 30091 + 12, 30109 = 30091 + 18 and 30113 = 30091 + 22 are consecutive primes.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Subsequence of A033451. - R. J. Mathar, May 06 2017

Cf. A001223, A078866, A078867, A078946-A078971, A022006, A022007.

Select[Partition[Prime[Range[150000]], 5, 1], Differences[#] == {6,6,6,4} &][[;;, 1]] (* Amiram Eldar, Feb 22 2025 *)

list(lim) = {my(p1 = 2, p2 = 3, p3 = 5, p4 = 7); forprime(p5 = 11, lim, if(p2 - p1 == 6 && p3 - p2 == 6 && p4 - p3 == 6 && p5 - p4 == 4, print1(p1, ", ")); p1 = p2; p2 = p3; p3 = p4; p4 = p5);} \\ Amiram Eldar, Feb 22 2025

a(n) == 1 (mod 30). - Amiram Eldar, Feb 22 2025

Edited by Dean Hickerson, Dec 20 2002

A079016 Suppose p and q = p+12 are primes. Define the difference pattern of (p,q) to be the successive differences of the primes in the range p to q. There are 14 possible difference patterns, namely [12], [2,10], [4,8], [6,6], [8,4], [10,2], [2,4,6], [2,6,4], [4,2,6], [4,6,2], [6,2,4], [6,4,2], [2,4,2,4] and [4,2,4,2]. Sequence gives smallest value of p for each difference pattern, sorted by magnitude.

Original entry on oeis.org

5, 7, 17, 19, 29, 31, 47, 67, 89, 137, 139, 199, 397, 1601

Offset: 1

Labos Elemer, Jan 24 2003

			p=1601, q=1613 has difference pattern [6,2,4] and {1601,1607,1609,1613} is the corresponding consecutive prime 4-tuple.

A022006(1)=5, A022007(1)=7, A078847(1)=17, A078851(1)=19, A078848(1)=29, A078855(1)=31, A047948(1)=47, A078850(1)=67, A031930(1)=A000230(6)=199, A046137(1)=7, A078853(1)=1601.

Cf. A079017-A079024.

Function[s, Function[t, Union@ Flatten@ Map[s[[First@ Position[t, #]]] &, {{12}, {2, 10}, {4, 8}, {6, 6}, {8, 4}, {10, 2}, {2, 4, 6}, {2, 6, 4}, {4, 2, 6}, {4, 6, 2}, {6, 2, 4}, {6, 4, 2}, {2, 4, 2, 4}, {4, 2, 4, 2}}]]@ Map[Differences@ Select[Range[#, # + 12], PrimeQ] &, s]]@ Select[Prime@ Range[10^3], PrimeQ[# + 12] &] (* Michael De Vlieger, Feb 25 2017 *)

A078868 Decimal concatenations of the quadruples (d1,d2,d3,d4) with elements in {2,4,6} for which there exists a prime p >= 5 such that the differences between the 5 consecutive primes starting with p are (d1,d2,d3,d4).

Original entry on oeis.org

2424, 2462, 2466, 2642, 2646, 2664, 2666, 4242, 4246, 4264, 4624, 4626, 4662, 4666, 6246, 6264, 6266, 6424, 6426, 6462, 6466, 6626, 6642, 6646, 6662, 6664

Offset: 1

Labos Elemer, Dec 19 2002

			4624 corresponds to the quadruple (4,6,2,4). It is in the sequence because the 5 consecutive primes 1597, 1601, 1607, 1609 and 1613 have differences (4,6,2,4).

The least primes corresponding to the quadruples are in A078866. The same primes, in increasing order, are in A078867. The sequences of primes corresponding to the 26 difference patterns are in A022006 (for 2424), A022007 (for 4242) and A078946-A078970. The similarly defined quintuples are in A078870. Cf. A001223.

With[{k = 4}, FromDigits /@ Select[Tuples[Range[2, 6, 2], k], Function[m, Count[Range[k, 10^k], n_ /; Times @@ Boole@ Map[PrimeQ, Prime@ n + Accumulate@ m] == 1] > 0]]] (* Michael De Vlieger, Mar 25 2017 *) (* or *)
FromDigits /@ Union@ Select[ Partition[ Differences@ Prime@ Range[3, 2000], 4, 1], Max@ # <= 6 &] (* Giovanni Resta, Mar 25 2017 *)

Edited by Dean Hickerson, Dec 20 2002

A186634 Irregular triangle, read by rows, giving dense patterns of n primes.

Original entry on oeis.org

0, 2, 0, 2, 6, 0, 4, 6, 0, 2, 6, 8, 0, 2, 6, 8, 12, 0, 4, 6, 10, 12, 0, 4, 6, 10, 12, 16, 0, 2, 6, 8, 12, 18, 20, 0, 2, 8, 12, 14, 18, 20, 0, 2, 6, 8, 12, 18, 20, 26, 0, 2, 6, 12, 14, 20, 24, 26, 0, 6, 8, 14, 18, 20, 24, 26, 0, 2, 6, 8, 12, 18, 20, 26, 30, 0, 2, 6, 12, 14, 20, 24, 26, 30, 0, 4, 6, 10, 16, 18, 24, 28, 30, 0, 4, 10, 12, 18, 22, 24, 28, 30, 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 0, 2, 6, 12, 14, 20, 24, 26, 30, 32

Offset: 2

T. D. Noe, Feb 24 2011

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.

			The irregular triangle begins:
0, 2
0, 2, 6, 0, 4, 6
0, 2, 6, 8
0, 2, 6, 8, 12, 0, 4, 6, 10, 12
0, 4, 6, 10, 12, 16
0, 2, 6, 8, 12, 18, 20, 0, 2, 8, 12, 14, 18, 20

T. D. Noe, Rows n = 2..20, flattened (from Forbes)
Thomas J. Engelsma, Permissible Patterns
Tony Forbes, Prime clusters and Cunningham chains, Math. Comp. 68 (1999), 1739-1747.
Tony Forbes, Smallest Prime k-tuples
Eric Weisstein's World of Mathematics, Prime Constellation

Cf. A020497, A008407, A186702, and sequences created by these patterns: A001359, A022004, A022005, A007530, A022006, A022007, A022008, A022009, A022010, A022011, A022012, A022013, A022545, A022546, A022547, A022548, A027569, A027570.

cp's OEIS Frontend

A257374 Numbers n such that n, n+4, n+10, n+12, n+16, n+22, n+24, n+30, n+36, n+40, n+42, n+46, n+52, n+54, n+60, n+64 and n+66 are all prime.

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A257375 Numbers n such that n, n+4, n+6, n+10, n+16, n+18, n+24, n+28, n+30, n+34, n+40, n+46, n+48, n+54, n+58, n+60 and n+66 are all prime.

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A257376 Numbers n such that n, n+6, n+8, n+12, n+18, n+20, n+26, n+32, n+36, n+38, n+42, n+48, n+50, n+56, n+60, n+62 and n+66 are all prime.

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A257377 Numbers n such that n, n+2, n+6, n+12, n+14, n+20, n+24, n+26, n+30, n+36, n+42, n+44, n+50, n+54, n+56, n+62 and n+66 are all prime.

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A201062 Record (maximal) gaps between prime 5-tuples (p, p+4, p+6, p+10, p+12).

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A201073 Record (maximal) gaps between prime 5-tuples (p, p+2, p+6, p+8, p+12).

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A078969 Primes p such that the differences between the 5 consecutive primes starting with p are (6,6,6,4).

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A078868 Decimal concatenations of the quadruples (d1,d2,d3,d4) with elements in {2,4,6} for which there exists a prime p >= 5 such that the differences between the 5 consecutive primes starting with p are (d1,d2,d3,d4).

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A186634 Irregular triangle, read by rows, giving dense patterns of n primes.

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