A200503 -id:A200503 - cp's OEIS frontend

A200504 Initial primes in prime sextuplets (p, p+4, p+6, p+10, p+12, p+16) preceding the maximal gaps in A200503.

7, 97, 19417, 43777, 3400207, 11664547, 37055647, 82984537, 89483827, 94752727, 381674467, 1569747997, 2019957337, 5892947647, 6797589427, 14048370097, 23438578897, 24649559647, 29637700987, 29869155847, 45555183127, 52993564567, 58430706067, 93378527647

Offset: 1

Alexei Kourbatov, Nov 18 2011

Prime sextuplets (p, p+4, p+6, p+10, p+12, p+16) are densest permissible constellations of 6 primes. The maximal gaps between prime sextuplets are listed in A200503; see further comments there.

			Two smallest prime sextuplets (p, p+4, p+6, p+10, p+12, p+16) start at p=7 and p=97; so a(1)=7. The gap of 15960 between sextuplets starting at p=97 and p=16057 is a maximal gap - larger than any preceding gap; so a(2)=97. The next gap is smaller, so 16057 is not in A200504. The gap of 24360 after the sextuplet starting at p=19417 is a maximal gap, therefore a(3)=19417; and so on.

Alexei Kourbatov, Table of n, a(n) for n = 1..56
Tony Forbes, List of all possible patterns of prime k-tuplets (up to k=50)
Alexei Kourbatov, Maximal gaps between prime k-tuples
Alexei Kourbatov and Marek Wolf, Predicting maximal gaps in sets of primes, arXiv preprint arXiv:1901.03785 [math.NT], 2019.

Cf. A022008 (prime sextuplets), A200503, A233426.

A233426 Primes p in prime sextuplets (p, p+4, p+6, p+10, p+12, p+16) at the end of the maximal gaps in A200503.

Original entry on oeis.org

97, 16057, 43777, 1091257, 6005887, 14520547, 40660717, 87423097, 94752727, 112710877, 403629757, 1593658597, 2057241997, 5933145847, 6860027887, 14112464617, 23504713147, 24720149677, 29715350377, 29952516817, 45645253597, 53086708387, 58528934197, 93495691687, 97367556817

Offset: 1

Alexei Kourbatov, Dec 09 2013

nonn

Prime sextuplets (p, p+4, p+6, p+10, p+12, p+16) are densest permissible constellations of 6 primes. Maximal (record) gaps between prime sextuplets are listed in A200503; see further comments there.

			Two smallest prime sextuplets (p, p+4, p+6, p+10, p+12, p+16) start at p=7 and p=97; so a[1]=97. The gap of 15960 between sextuplets starting at p=97 and p=16057 is a record gap - larger than any preceding gap; so a[2]=16057. The next gap is not a record, so a new term is not added.

Alexei Kourbatov, Table of n, a(n) for n = 1..56
Tony Forbes and Norman Luhn, Prime k-tuplets
Alexei Kourbatov, Maximal gaps between prime k-tuples
Alexei Kourbatov, Tables of record gaps between prime constellations, arXiv preprint arXiv:1309.4053, 2013.
Norman Luhn, Record Gaps Between Prime Sextuplets
Eric W. Weisstein, k-Tuple Conjecture

Cf. A022008, A200503, A200504.

a(n) = A200504(n) + A200503(n). - Hugo Pfoertner, May 21 2023

A202281 Record (maximal) gaps between prime decuplets (p+0,2,6,8,12,18,20,26,30,32).

Original entry on oeis.org

33081664140, 50040961320, 211797665730, 278538937950, 314694286830, 446820068310, 589320949140, 1135263664920, 1154348695500, 1280949740070, 1340804150070, 1458168320490, 1539906870810, 1858581264540, 2590180927950, 3182865274050, 4949076176310, 5719502339670

Offset: 1

Alexei Kourbatov, Dec 15 2011

nonn

Prime decuplets (p+0,2,6,8,12,18,20,26,30,32) are one of the two types of densest permissible constellations of 10 primes (A027569 and A027570).
Average gaps between prime k-tuples are O(log^k(p)), with k=10 for decuplets, by the Hardy-Littlewood k-tuple conjecture. 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^11(p)).
A202282 lists initial primes in decuplets (p+0,2,6,8,12,18,20,26,30,32) preceding the maximal gaps.

			The gap of 33081664140 after the first decuplet starting at p=11 is the term a(1). The next three gaps of 50040961320, 211797665730, 278538937950 form an increasing sequence, each setting a new record; therefore each of these gaps is in the sequence, as a(2), a(3), a(4). The next gap is not a record, so it is not in this sequence.

Norman Luhn, Table of n, a(n) for n = 1..54 (terms 1..32 from Dana Jacobsen).
Tony Forbes and Norman Luhn, 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 Math. 44, 1-70, 1923.
Alexei Kourbatov, Maximal gaps between prime k-tuples
Alexei 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.
Eric Weisstein's World of Mathematics, k-Tuple Conjecture

Cf. A027569 (prime decuplets p+0,2,6,8,12,18,20,26,30,32), A202282, A202361, A113274, A113404, A200503, A201596, A201598, A201062, A201073, A201051, A201251

use ntheory ":all"; my($i,$l,$max)=(-1,0,0); for (sieve_prime_cluster(1,1e13,2,6,8,12,18,20,26,30,32)) { my $gap=$-$l; if ($gap>$max) { say "$i $gap" if ++$i > 0; $max=$gap; } $l=$; } # Dana Jacobsen, Oct 08 2015

(1) Upper bound: gaps between prime decuplets (p+0,2,6,8,12,18,20,26,30,32) are smaller than 0.00059*(log p)^11, where p is the prime at the end of the gap.
(2) Estimate for the actual size of maximal gaps near p: max gap = a(log(p/a)-0.2), where a = 0.00059(log p)^10 is the average gap between 10-tuples near p.
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.

A202361 Record (maximal) gaps between prime decuplets (p+0,2,6,12,14,20,24,26,30,32).

Original entry on oeis.org

12102794130, 141702673770, 424052301750, 699699330330, 714303547230, 739544215410, 1623198312120, 2691533434590, 4207848555330, 4936074819480, 5887574660310, 6562654104930, 7205070907650, 8129061524010, 8362548652500, 9741706748970, 9967327212570

Offset: 1

Alexei Kourbatov, Dec 18 2011

nonn

Prime decuplets (p+0,2,6,12,14,20,24,26,30,32) are one of the two types of densest permissible constellations of 10 primes (A027569 and A027570).
Average gaps between prime k-tuples are O(log^k(p)), with k=10 for decuplets, by the Hardy-Littlewood k-tuple conjecture. 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^11(p)).
A202362 lists initial primes in decuplets (p+0,2,6,12,14,20,24,26,30,32) preceding the maximal gaps.

			The gap of 12102794130 between the very first decuplets starting at p=9853497737 and p=21956291867 is the initial term a(1)=12102794130.
The next gap after the decuplet starting at p=21956291867 is smaller, so it is not in this sequence.
The next gap of 141702673770 between the decuplets at p=22741837817 and p=164444511587 is a new record; therefore the next term is a(2)=141702673770.

Norman Luhn, Table of n, a(n) for n = 1..49 (terms 1..27 from Dana Jacobsen).
Tony Forbes and Norman Luhn, 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 Math. 44, 1-70, 1923.
Alexei Kourbatov, Maximal gaps between prime k-tuples
Alexei 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.
Eric Weisstein's World of Mathematics, k-Tuple Conjecture

Cf. A027570 (prime decuplets p+0,2,6,12,14,20,24,26,30,32), A202362, A113274, A113404, A200503, A201596, A201598, A201062, A201073, A201051, A201251, A202281.

use ntheory ":all"; my($i,$l,$max)=(-1,0,0); for (sieve_prime_cluster(1,1e13,2,6,12,14,20,24,26,30,32)) { my $gap=$-$l; if ($gap>$max) { say "$i $gap" if ++$i > 0; $max=$gap; } $l=$; } # Dana Jacobsen, Oct 09 2015

(1) Upper bound: gaps between prime decuplets (p+0,2,6,12,14,20,24,26,30,32) are smaller than 0.00059*(log p)^11, where p is the prime at the end of the gap.
(2) Estimate for the actual size of maximal gaps near p: max gap = a(log(p/a)-0.2), where a = 0.00059*(log p)^10 is the average gap between 10-tuples near p.
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.

A201051 Record (maximal) gaps between prime septuplets (p, p+2, p+6, p+8, p+12, p+18, p+20).

Original entry on oeis.org

165690, 903000, 10831800, 13773480, 22813770, 31090080, 43751820, 60881310, 86746170, 118516860, 239951250, 281573040, 359932650, 384903750, 518385000, 902801550, 1027007520, 1086331680, 1329198570, 2176467090

Offset: 1

Alexei Kourbatov, Nov 28 2011

Prime septuplets (p, p+2, p+6, p+8, p+12, p+18, p+20) are one of the two types of densest permissible constellations of 7 primes (A022009 and A022010). Average gaps between prime k-tuples can be deduced from the Hardy-Littlewood k-tuple conjecture and are O(log^k(p)), with k=7 for septuplets. 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^8(p)).

A201249 lists initial primes p in septuplets (p, p+2, p+6, p+8, p+12, p+18, p+20) preceding the maximal gaps. A233425 lists the corresponding primes at the end of the maximal gaps.

			The gap of 165690 between septuplets starting at p=11 and p=165701 is the very first gap, so a(1)=165690. The gap of 903000 between septuplets starting at p=165701 and p=1068701 is a maximal gap - larger than any preceding gap; therefore a(2)=903000. The next gap of 10831800 is again a maximal gap, so a(3)=10831800. The next gap is smaller, so it does not contribute to the sequence.

Alexei Kourbatov, Table of n, a(n) for n = 1..36
Tony Forbes and Norman Luhn, 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 Math., Vol. 44, No. 1 (1923), pp. 1-70.
Alexei Kourbatov, Maximal gaps between prime k-tuples
Alexei 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.
Norman Luhn, Record Gaps Between Prime Septuplets, up to 10^17
Eric Weisstein's World of Mathematics, k-Tuple Conjecture

Cf. A022009 (prime septuplets p, p+2, p+6, p+8, p+12, p+18, p+20), A113274, A113404, A200503, A201062, A201073, A201596, A201598, A201251, A202281, A202361, A201249, A002386, A233425.

Gaps between prime septuplets (p, p+2, p+6, p+8, p+12, p+18, p+20) are smaller than 0.02*(log p)^8, where p is the prime at the end of the gap. There is no rigorous proof of this formula. The O(log^8(p)) growth rate is suggested by numerical data and heuristics based on probability considerations.

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...

A201251 Record (maximal) gaps between prime septuplets (p, p+2, p+8, p+12, p+14, p+18, p+20).

Original entry on oeis.org

83160, 195930, 341880, 5414220, 9270030, 18980220, 25622520, 36077370, 51597630, 92184750, 125523090, 140407470, 141896370, 336026460, 403369470, 435390270, 442452570, 627852330, 754383210, 1008582120, 1021464990, 1073692620, 1088148810, 1145336850

Offset: 1

Alexei Kourbatov, Nov 28 2011

Prime septuplets (p, p+2, p+8, p+12, p+14, p+18, p+20) are one of the two types of densest permissible constellations of 7 primes (A022009 and A022010). Average gaps between prime k-tuples can be deduced from the Hardy-Littlewood k-tuple conjecture and are O(log^k(p)), with k=7 for septuplets. 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^8(p)).

A201252 lists initial primes in septuplets (p, p+2, p+8, p+12, p+14, p+18, p+20) preceding the maximal gaps. A233038 lists the corresponding primes at the end of the maximal gaps.

			The gap of 83160 between septuplets starting at p=5639 and p=88799 is the very first gap, so a(1)=83160. The gap of 195930 between septuplets starting at p=88799 and p=284729 is a maximal gap - larger than any preceding gap; therefore a(2)=195930. The next gap of 341880 is again a maximal gap, so a(3)=341880. The next gap is smaller, so it does not contribute to the sequence.

Alexei Kourbatov, Table of n, a(n) for n = 1..52
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 Math., Vol. 44, No. 1 (1923), pp. 1-70.
Alexei Kourbatov, Maximal gaps between prime k-tuples.
Alexei 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.
Norman Luhn, Patterns of prime k-tuplets & the Hardy-Littlewood constants.
Norman Luhn, Record Gaps Between Prime Septuplets, up to 10^17.
Eric Weisstein's World of Mathematics, k-Tuple Conjecture.

Cf. A022010 (prime septuplets p, p+2, p+8, p+12, p+14, p+18, p+20), A113274, A113404, A200503, A201062, A201073, A201596, A201598, A202281, A202361, A201051, A002386, A233038.

Gaps between prime septuplets (p, p+2, p+8, p+12, p+14, p+18, p+20) are smaller than 0.02*(log p)^8, where p is the prime at the end of the gap. There is no rigorous proof of this formula. The O(log^8(p)) growth rate is suggested by numerical data and heuristics based on probability considerations.

A201596 Record (maximal) gaps between prime triples (p, p+4, p+6).

Original entry on oeis.org

6, 24, 30, 90, 150, 156, 210, 240, 306, 366, 384, 444, 810, 834, 1086, 1200, 1326, 2316, 3876, 4230, 4350, 8244, 8880, 9450, 10686, 10950, 11784, 12816, 13554, 15504, 15576, 16254, 16506, 16596, 19446, 19944, 21516, 38340, 39990, 41556, 45786, 47190, 48246, 59856

Offset: 1

Alexei Kourbatov, Dec 03 2011

nonn

Prime triples (p, p+4, p+6) are one of the two types of densest permissible constellations of 3 primes (A022004 and A022005). By the Hardy-Littlewood k-tuple conjecture, average gaps between prime k-tuples are O(log^k(p)), with k=3 for triples. 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 between triples are O(log^4(p)).

A201597 lists initial primes p in triples (p, p+4, p+6) preceding the maximal gaps. A233435 lists the corresponding primes p at the end of the maximal gaps.

			The gap of 6 between triples starting at p=7 and p=13 is the very first gap, so a(1)=6. The gap of 24 between triples starting at p=13 and p=37 is a maximal gap - larger than any preceding gap; therefore a(2)=24. The gap of 30 between triples at p=37 and p=67 is again a maximal gap, so a(3)=30. The next gap is smaller, so it does not contribute to the sequence.

Alexei Kourbatov, Table of n, a(n) for n = 1..79
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 k-tuples
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.
Norman Luhn, Record Gaps Between Prime Triplets.
Eric W. Weisstein, k-Tuple Conjecture

Cf. A022005 (prime triples p, p+4, p+6), A113274, A113404, A200503, A201598, A201062, A201073, A201051, A201251, A202281, A202361, A201597, A233435.

DeleteDuplicates[Differences[Select[Partition[Prime[Range[5*10^6]],3,1],Differences[#]=={4,2}&][[;;,1]]],GreaterEqual]  (* Harvey P. Dale, Feb 26 2023 *)

Gaps between prime triples (p, p+4, p+6) are smaller than 0.35*(log p)^4, where p is the prime at the end of the gap. There is no rigorous proof of this formula. The O(log^4(p)) growth rate is suggested by numerical data and heuristics based on probability considerations.

A201598 Record (maximal) gaps between prime triples (p, p+2, p+6).

Original entry on oeis.org

6, 24, 60, 84, 114, 180, 210, 264, 390, 564, 630, 1050, 1200, 1530, 2016, 2844, 3426, 3756, 3864, 3936, 4074, 4110, 6090, 8250, 9240, 9270, 10344, 10506, 10734, 10920, 12930, 15204, 20190, 20286, 21216, 25746, 34920, 38820, 39390, 41754, 43020, 44310, 52500, 71346

Offset: 1

Alexei Kourbatov, Dec 03 2011

nonn

Prime triples (p, p+2, p+6) are one of the two types of densest permissible constellations of 3 primes (A022004 and A022005). By the Hardy-Littlewood k-tuple conjecture, average gaps between prime k-tuples are O(log^k(p)), with k=3 for triples. 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 between triples are O(log^4(p)).

A201599 lists initial primes p in triples (p, p+2, p+6) preceding the maximal gaps. A233434 lists the corresponding primes p at the end of the maximal gaps.

			The gap of 6 between triples starting at p=5 and p=11 is the very first gap, so a(1)=6. The gap of 6 between triples starting at p=11 and p=17 is not a record, so it does not contribute to the sequence. The gap of 24 between triples starting at p=17 and p=41 is a maximal gap - larger than any preceding gap; therefore a(2)=24.

Alexei Kourbatov, Table of n, a(n) for n = 1..72
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 k-tuples
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.
Norman Luhn, Record Gaps Between Prime Triplets.
Eric Weisstein's World of Mathematics, k-Tuple Conjecture.

Cf. A022004 (prime triples p, p+2, p+6), A113274, A113404, A200503, A201596, A201062, A201073, A201051, A201251, A202281, A202361, A201599, A233434.

Gaps between prime triples (p, p+2, p+6) are smaller than 0.35*(log p)^4, where p is the prime at the end of the gap. There is no rigorous proof of this formula. The O(log^4(p)) growth rate is suggested by numerical data and heuristics based on probability considerations.

cp's OEIS Frontend

A200504 Initial primes in prime sextuplets (p, p+4, p+6, p+10, p+12, p+16) preceding the maximal gaps in A200503.

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A233426 Primes p in prime sextuplets (p, p+4, p+6, p+10, p+12, p+16) at the end of the maximal gaps in A200503.

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A202281 Record (maximal) gaps between prime decuplets (p+0,2,6,8,12,18,20,26,30,32).

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A202361 Record (maximal) gaps between prime decuplets (p+0,2,6,12,14,20,24,26,30,32).

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Perl

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A201051 Record (maximal) gaps between prime septuplets (p, p+2, p+6, p+8, p+12, p+18, p+20).

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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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Mathematica

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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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A201251 Record (maximal) gaps between prime septuplets (p, p+2, p+8, p+12, p+14, p+18, p+20).

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A201596 Record (maximal) gaps between prime triples (p, p+4, p+6).

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