A022004 -id:A022004 - 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.

A190817 Initial primes of 6 consecutive primes with consecutive gaps 2,4,6,8,10.

Original entry on oeis.org

13901, 21557, 28277, 55661, 68897, 128981, 221717, 354371, 548831, 665111, 954257, 1164587, 1246367, 1265081, 1538081, 1595051, 1634441, 2200811, 2798921, 2858621, 3053747, 3407081, 3414011, 3967487, 3992201, 4480241, 4608281, 4701731, 4809251, 5029457

Offset: 1

Zak Seidov, May 21 2011

nonn

a(1) = 13901 = A190814(5) = A187058(7) = A078847(24).

a(n) + 30 is the greatest term in the sequence of 6 consecutive primes with consecutive gaps 2, 4, 6, 8, 10. - Muniru A Asiru, Aug 10 2017

			For n = 1, 13901 is in the sequence because 13901, 13903, 13907, 13913, 13921, 13931 are consecutive primes and for n = 2, 21557 is in the sequence since 21557, 21559, 21563, 21569, 21577, 21587 are consecutive primes. - _Muniru A Asiru_, Aug 24 2017

Zak Seidov, Table of n, a(n) for n = 1..6000
R. J. Mathar, Table of Prime Gap Constellations

Cf. A078847, A187058, A190814, A190819, A190838.

K:=3*10^7+1;; # to get all terms <= K.
P:=Filtered([1,3..K],IsPrime);; I:=[2,4,6,8,10];;
P1:=List([1..Length(P)-1],i->P[i+1]-P[i]);;
P2:=List([1..Length(P)-Length(I)],i->[P1[i],P1[i+1],P1[i+2],P1[i+3],P1[i+4]]);;
P3:=List(Positions(P2,I),i->P[i]);  # Muniru A Asiru, Aug 24 2017

N:=10^7: # to get all terms <= N.
Primes:=select(isprime,[seq(i,i=3..N+30,2)]):
Primes[select(t->[Primes[t+1]-Primes[t], Primes[t+2]-Primes[t+1],Primes[t+3]-Primes[t+2], Primes[t+4]-Primes[t+3], Primes[t+5]-Primes[t+4]]=[2,4,6,8,10], [$1..nops(Primes)-5])]; # Muniru A Asiru, Aug 04 2017

d = Differences[Prime[Range[100000]]]; Prime[Flatten[Position[Partition[d, 5, 1], {2, 4, 6, 8, 10}]]] (* T. D. Noe, May 23 2011 *)
With[{s = Differences@ Prime@ Range[10^6]}, Prime[SequencePosition[s, Range[2, 10, 2]][[All, 1]] ] ] (* Michael De Vlieger, Aug 16 2017 *)

lista(nn) = forprime(p=13901, nn, if(nextprime(p+1)==p+2 && nextprime(p+3)==p+6 && nextprime(p+7)==p+12 && nextprime(p+13)==p+20 && nextprime(p+21)==p+30, print1(p", "))); \\ Altug Alkan, Aug 16 2017

Additional cross references from Harvey P. Dale, May 10 2014

A191456 Primes p such that the polynomial x^2+x+p generates only primes for x=1..9.

Original entry on oeis.org

11, 17, 41, 844427, 51448361, 86966771, 122983031, 180078317, 960959381, 1278189947, 1761702947, 1829187287, 2426256797, 2911675511, 3013107257, 4778888351, 5221343711

Offset: 1

Zak Seidov, Jun 02 2011

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..2713

Generates primes for x=1..k: A001359 (1), A022004 (2), A172454 (3), A187057 (4), A187058 (5), A144051 (6), A187060 (7), A190800 (8), this sequence (9), A191457 (10), A191458 (11), A253592 (12), A253605 (13). Each is by definition a subsequence of preceding sequences.
Subsequence such that x=10 gives a composite number: A211238.
Cf. A160548, A164926.

is(n)=for(x=0,9, if(!isprime(x^2+x+n), return(0))); 1 \\ Charles R Greathouse IV, Sep 14 2015

A073648 Middle members of prime triples {p, p+2, p+6}.

Original entry on oeis.org

7, 13, 19, 43, 103, 109, 193, 229, 313, 349, 463, 643, 823, 859, 883, 1093, 1279, 1303, 1429, 1483, 1489, 1609, 1873, 1999, 2083, 2239, 2269, 2659, 2689, 3253, 3463, 3529, 3673, 3919, 4003, 4129, 4519, 4639, 4789, 4933, 4969, 5233, 5479, 5503, 5653, 6199

Offset: 1

Amarnath Murthy, Aug 09 2002

Zak Seidov, Table of n,a(n) for n=1,2380, a(n)<2*10^6

Cf. A073649, A073650.

Cf. A098412, A098414.

Transpose[Select[Partition[Prime[Range[850]],3,1],Differences[#]=={2,4}&]][[2]]  (* Harvey P. Dale, Feb 20 2011 *)

a(n) = A022004(n) + 2.

More terms from Benoit Cloitre, Aug 13 2002

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.

A098412 Greatest members p of prime triples (p-6, p-4, p).

Original entry on oeis.org

11, 17, 23, 47, 107, 113, 197, 233, 317, 353, 467, 647, 827, 863, 887, 1097, 1283, 1307, 1433, 1487, 1493, 1613, 1877, 2003, 2087, 2243, 2273, 2663, 2693, 3257, 3467, 3533, 3677, 3923, 4007, 4133, 4523, 4643, 4793, 4937, 4973, 5237, 5483, 5507, 5657, 6203

Offset: 1

Reinhard Zumkeller, Sep 07 2004

nonn

Subsequence of A046117; a(n) = A073648(n) + 4 = A022004(n) + 6.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Prime Triplet

Cf. A098413, A098415.

[p: p in PrimesUpTo(6500)|IsPrime(p) and IsPrime(p-6) and IsPrime(p-4)]; // Vincenzo Librandi, Dec 26 2010

K:=10^7: # to get all terms <= K.
for n from 1 by 2 to K do; if isprime(n-6) and isprime(n-4) and isprime(n) then print(n) else fi; od;  # Muniru A Asiru, Aug 06 2017

Select[Table[Prime[n], {n, 1000}], PrimeQ[# - 4] && PrimeQ[# - 6] &] (* Vladimir Joseph Stephan Orlovsky, Jun 19 2011 *)
Select[Partition[Prime[Range[1000]],3,1],Differences[#]=={2,4}&][[All,3]] (* Harvey P. Dale, Sep 23 2017 *)

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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A190817 Initial primes of 6 consecutive primes with consecutive gaps 2,4,6,8,10.

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GAP

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A191456 Primes p such that the polynomial x^2+x+p generates only primes for x=1..9.

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PARI

A073648 Middle members of prime triples {p, p+2, p+6}.

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

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Mathematica

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

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A098412 Greatest members p of prime triples (p-6, p-4, p).