A022007 -id:A022007 - cp's OEIS frontend

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.

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.

A096502 a(n) = k is the smallest exponent k such that 2^k - (2n+1) is a prime number, or 0 if no such k exists.

Original entry on oeis.org

2, 3, 3, 39, 4, 4, 4, 5, 6, 5, 5, 6, 5, 5, 5, 7, 6, 6, 11, 7, 6, 29, 6, 6, 7, 6, 6, 7, 6, 6, 6, 8, 8, 7, 7, 10, 9, 7, 8, 9, 7, 8, 7, 7, 8, 7, 8, 10, 7, 7, 26, 9, 7, 8, 7, 7, 10, 7, 7, 8, 7, 7, 7, 47, 8, 14, 9, 11, 10, 9, 10, 8, 9, 8, 8, 31, 8, 8, 15, 8, 10, 9

Offset: 0

Labos Elemer, Jul 09 2004

As D. W. Wilson observes, this is similar to the Riesel/Sierpinski problem and there is e.g. no prime of the form 2^k - 777149, which is divisible by 3,5,7,13,19,37 or 73 if k is in 1+2Z, 2+4Z, 4+12Z, 8+12Z, 12+36Z, 0+36Z resp. 24+36Z. Already for n=935 it is difficult to find a solution. Is this linked to the fact that 2n+1=1871 is member of a prime quadruple (A007530) and quintuple (A022007)? - M. F. Hasler, Apr 07 2008

			a(0)=A000043(1)=2, a(1)=A050414(1)=3, a(2)=A059608(1)=3, a(3)=A059609(1)=39.
For n=110 and n=111 even these smallest exponents are rather large: a(110)=714, a(111)=261 which mean that 2^714-221 and 2^261-223 are the least corresponding prime numbers.

T. D. Noe, Table of n, a(n) for n = 0..934
F. Firoozbakht, M. F. Hasler, Variations on Euclid's formula for Perfect Numbers, JIS 13 (2010) #10.3.1.

Cf. A000043, A007530, A022007, A050414, A059608, A059609.

Table[k = 1; While[2^k < n || ! PrimeQ[2^k - n], k++]; k, {n, 1, 1869, 2}] (* T. D. Noe, Mar 18 2013 *)

A096502(n,k)={ k || k=log(n)\log(2)+1; n=2*n+1; while( !ispseudoprime(2^k++-n),);k } /* will take a long time for n=935... */ - M. F. Hasler, Apr 07 2008

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

cp's OEIS Frontend

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.

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A257373 Initial members of prime 17-tuplets.

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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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A096502 a(n) = k is the smallest exponent k such that 2^k - (2n+1) is a prime number, or 0 if no such k exists.

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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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