A027569 -id:A027569 - cp's OEIS frontend

A257307 Numbers n such that n, n+6, n+8, n+14, n+20, n+24, n+26, n+30, n+36, n+38, n+44, n+48, n+50, n+54 and n+56 are all prime.

14094050870111867483, 16222575536498135633, 27367669288651556693, 47972909762951107763, 49198349218890756713, 83522578855892290943, 141851207652098108993, 166270537650069307283, 195690251815182929633, 334275099381872183363, 347599197380365425323

Offset: 1

Tim Johannes Ohrtmann, Apr 20 2015

nonn

Dana Jacobsen, Table of n, a(n) for n = 1..53 [first 9 terms computed by Jens Kruse Andersen and Jörg Waldvogel (2007-2009)]
Tony Forbes and Norman Luhn, Smallest Prime k-tuplets
Norman Luhn, Table of n, a(n) for n = 1..68 (up to 10^22)

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, this sequence.
prime 16-tuplets: A257308 out of A257369, A257370.
prime 17-tuplets: A257373 out of A257374, A257375, A257376, A257377.

Corrected a(2) and more terms from Dana Jacobsen, Oct 31 2015

A257308 Initial members of prime 16-tuplets.

Original entry on oeis.org

13, 47710850533373130107, 347709450746519734877, 695874886175252911063, 1099638576123052218257, 1169914227530138703617, 1522014304823128379267, 1567582627835236839763, 1620784518619319025977, 1750052554011927712483, 2257588388550898970503, 2639154464612254121537, 3259125690557440336637, 3789227751026345304613, 4654682384109074514133, 5022156579757255625623, 9042634271485192050677, 9239395687646993061197, 13599236099159166553033

Offset: 1

Tim Johannes Ohrtmann, Apr 21 2015

nonn

Tim Johannes Ohrtmann, Table of n, a(n) for n = 1..66
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: this sequence out of A257369, A257370.
prime 17-tuplets: A257373 out of A257374, A257375, A257376, A257377.

A257369 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 and n+60 are all prime.

Original entry on oeis.org

13, 695874886175252911063, 1567582627835236839763, 1750052554011927712483, 2257588388550898970503, 3789227751026345304613, 4654682384109074514133, 5022156579757255625623, 13599236099159166553033, 29894522822363684652103, 35718904544536715448883, 42421183685552747462323, 47624415490498763963983, 50069823850049036630533, 56294926786180569503953, 60877851518090858117803, 66871135379953148611303, 70743491366529526461853, 72039555441202354852783

Offset: 1

Tim Johannes Ohrtmann, Apr 21 2015

nonn

Tim Johannes Ohrtmann, Table of n, a(n) for n = 1..30
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 this sequence, A257370.
prime 17-tuplets: A257373 out of A257374, A257375, A257376, A257377.

A257370 Numbers n such that n, n+2, n+6, n+12, n+14, n+20, n+26, n+30, n+32, n+36, n+42, n+44, n+50, n+54, n+56 and n+60 are all prime.

Original entry on oeis.org

47710850533373130107, 347709450746519734877, 1099638576123052218257, 1169914227530138703617, 1522014304823128379267, 1620784518619319025977, 2639154464612254121537, 3259125690557440336637, 9042634271485192050677, 9239395687646993061197, 15571053758048293307807, 20628149050698694668167, 20947353617877810296177, 23182160505954925788317, 27814116054901200587567, 30406149669349341460577, 31607383424682394081757, 34254730511961158822627

Offset: 1

Tim Johannes Ohrtmann, Apr 21 2015

nonn

Tim Johannes Ohrtmann, Table of n, a(n) for n = 1..37
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, this sequence.
prime 17-tuplets: A257373 out of A257374, A257375, A257376, A257377.

A257373 Initial members of prime 17-tuplets.

Original entry on oeis.org

13, 17, 1620784518619319025971, 2639154464612254121531, 3259125690557440336631, 37630850994954402655487, 47624415490498763963983, 53947453971035573715707, 78314167738064529047713, 83405687980406998933663, 110885131130067570042703, 124211857692162527019731

Offset: 1

Tim Johannes Ohrtmann, Apr 21 2015

nonn

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: this sequence out of A257374, A257375, A257376, A257377.

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.

Original entry on oeis.org

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.

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.

cp's OEIS Frontend

A257307 Numbers n such that n, n+6, n+8, n+14, n+20, n+24, n+26, n+30, n+36, n+38, n+44, n+48, n+50, n+54 and n+56 are all prime.

Views

Author

Keywords

Links

Crossrefs

Extensions

A257308 Initial members of prime 16-tuplets.

Views

Author

Keywords

Links

Crossrefs

A257369 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 and n+60 are all prime.

Views

Author

Keywords

Links

Crossrefs

A257370 Numbers n such that n, n+2, n+6, n+12, n+14, n+20, n+26, n+30, n+32, n+36, n+42, n+44, n+50, n+54, n+56 and n+60 are all prime.

Views

Author

Keywords

Links

Crossrefs

A257373 Initial members of prime 17-tuplets.

Views

Author

Keywords

Links

Crossrefs

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.

Views

Author

Keywords

Links

Crossrefs

Extensions

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.

Views

Author

Keywords

Links

Crossrefs

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.

Views

Author

Keywords

Links

Crossrefs

Extensions

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.

Views

Author

Keywords

Links

Crossrefs

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Perl

Formula