A022007 -id:A022007 - cp's OEIS frontend

A375264 Products of prime 5-tuples (p, p+4, p+6, p+10, p+12) where p = A022007(n).

323323, 11769028333, 23075277662228863, 498323779650735373, 5774946339890457283, 964079483344036314403, 1069516017850492496773, 2764561257197073595813, 160896800504464381875853, 3214487754103795836790843, 5525765859191774942946523, 10569235342244010823461223, 1140114641548139662904736973

Offset: 1

Michael De Vlieger, Aug 16 2024

nonn

Subsequence of A046387.

All terms are congruent to 13 (mod 30), since they are products m of the following residues (mod 30): {7, 11, 13, 17, 19}, and m mod 30 is congruent to 13.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A022007, A046387, A375264.

Map[Times @@ NextPrime[#, Range[0, 4]] &, Select[Prime@ Range[2^16], AllTrue[{# + 4, # + 6, # + 10, # + 12}, PrimeQ] &]]

A199004 Position of n-th prime sextuplet A022008(n) among prime 5-tuples A022007.

Original entry on oeis.org

1, 2, 7, 8, 9, 35, 49, 55, 75, 76, 82, 84, 115, 124, 133, 140, 144, 153, 167, 172, 177, 200, 201, 202, 203, 211, 213, 232, 235, 237, 240, 243, 245, 252, 256, 262, 276, 281, 285, 297, 305, 319, 322, 325, 331, 349, 352, 353, 374, 377, 379, 380, 399, 407, 415

Offset: 1

Zak Seidov, Nov 01 2011

nonn

			a(3)=7 because A022008(3)=16057=A022007(7),
a(1000)=12091 because A022008(1000)=5054057077=A022007(12091).

Zak Seidov, Table of n, a(n) for n = 1..1000

Cf. A022007, A022008.

lst = {}; cnt = 0; Do[p = Prime[n]; If[PrimeQ[p+4] && PrimeQ[p+6] && PrimeQ[p+10] && PrimeQ[p+12], cnt++; If[PrimeQ[p+16], AppendTo[lst, cnt]]], {n, 10000000}]; lst (* T. D. Noe, Nov 01 2011 *)

A022007(a(n))=A022008(n).

A144940 Primes of form: sum of prime 5-tuples (A022007).

Original entry on oeis.org

67, 17317, 80317, 97117, 398467, 1333417, 1423717, 1876267, 2317267, 3332167, 4890367, 5456317, 5911417, 6151867, 6622867, 9151717, 9522367, 11013967, 13372717, 15381967, 17080267, 21663067, 22439017, 22552417, 24262267

Offset: 1

Vladimir Joseph Stephan Orlovsky, Sep 25 2008

nonn

lst={};Do[p=Prime[n];If[PrimeQ[p+4]&&PrimeQ[p+6]&&PrimeQ[p+10]&&PrimeQ[p+12],s=p+p+4+p+6+p+10+p+12;If[PrimeQ[s],AppendTo[lst,s]]],{n,10!}];lst
Select[Total/@Select[Partition[Prime[Range[350000]],5,1],Differences[#]=={4,2,4,2}&],PrimeQ] (* Harvey P. Dale, Dec 20 2022 *)

7+11+13+17+19=67

A022006 Initial members p of prime 5-tuples (p, p+2, p+6, p+8, p+12).

Original entry on oeis.org

5, 11, 101, 1481, 16061, 19421, 21011, 22271, 43781, 55331, 144161, 165701, 166841, 195731, 201821, 225341, 247601, 268811, 326141, 347981, 361211, 397751, 465161, 518801, 536441, 633461, 633791, 661091, 768191, 795791, 829721, 857951, 876011, 958541

Offset: 1

Warut Roonguthai

nonn

Subsequence of A007530. - R. J. Mathar, Feb 10 2013
All terms, except for the first one, are congruent to 11 (modulo 30). - Matt C. Anderson, May 22 2015
For n > 1 and p = a(n),  (p, p+2, p+6, p+8, p+12) are consecutive primes. - Zak Seidov, Jun 07 2017
A022007 is a similar sequence. - Wolfdieter Lang, Oct 06 2017

			Admissible 5-tuple guaranteeing sequence example: for prime(3) = 5 the first residue class starting with a nonnegative number and containing none of the members of (0, 2, 6, 8, 12) is 4 (mod 5). - _Wolfdieter Lang_, Oct 06 2017

Dana Jacobsen, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe, terms 1001..10000 from Matt C. Anderson)
J. K. Andersen, Prime Records.
Tony Forbes and Norman Luhn, Prime k-tuplets.
Norman Luhn, Table of n, a(n) for n = 1..1000000.

Cf. A000040, A022007.

[p: p in PrimesUpTo(2*10^6) | IsPrime(p+2) and IsPrime(p+6) and IsPrime(p+8) and IsPrime(p+12)]; // Vincenzo Librandi, May 23 2015

lst={};Do[p=Prime[n];If[PrimeQ[p+2]&&PrimeQ[p+6]&&PrimeQ[p+8]&&PrimeQ[p+12], AppendTo[lst, p]], {n, 9!}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 25 2008 *)
Transpose[Select[Partition[Prime[Range[64000]],5,1],Differences[#] == {2,4,2,4}&]][[1]] (* Harvey P. Dale, Dec 08 2014 *)

forprime(p=2,1e7, if(isprime(p+2) && isprime(p+6) && isprime(p+8) && isprime(p+12), print1(p", "))) \\ Charles R Greathouse IV, Jul 19 2011

use ntheory ":all"; say for sieve_prime_cluster(1,1e7, 2,6,8,12); # Dana Jacobsen, Sep 30 2015

from sympy import primerange
def aupto(limit):
  p, q, r, s, alst = 2, 3, 5, 7, []
  for t in primerange(11, limit+13):
    if p+2 == q and p+6 == r and p+8 == s and p+12 == t: alst.append(p)
    p, q, r, s = q, r, s, t
  return alst
print(aupto(10**6)) # Michael S. Branicky, May 11 2021

Missing terms a(51) and a(52) added in b-file by Dana Jacobsen, Sep 30 2015

A022008 Initial member of prime sextuples (p, p+4, p+6, p+10, p+12, p+16).

Original entry on oeis.org

7, 97, 16057, 19417, 43777, 1091257, 1615837, 1954357, 2822707, 2839927, 3243337, 3400207, 6005887, 6503587, 7187767, 7641367, 8061997, 8741137, 10526557, 11086837, 11664547, 14520547, 14812867, 14834707, 14856757, 16025827, 16094707, 18916477, 19197247

Offset: 1

Warut Roonguthai

nonn

Without the initial 7, this gives primes at which difference pattern X42424Y (X and Y >= 8) occurs in A001223. - Labos Elemer
Subsequence of A022007. - Zak Seidov, Nov 01 2011
From Jean-Christophe Hervé, Sep 27 2014: (Start)
The primes in a sextuple a(n), a(n)+4, a(n)+6, a(n)+10, a(n)+12, a(n)+16 are consecutive since a(n)+2, a(n)+8 and a(n)+14 cannot be prime (multiple of 3).
The prime sextuples starting at a(n) give the highest concentration of primes that can occur on an interval of 17 integers (apart intervals starting at p < 7). It is conjectured that there are infinitely many such sextuples.
For n > 1, the 3 odd integers preceding and the 3 odd integers following the sextuple are not prime: a(n)-2 == a(n)+18 == 0 (mod 5), a(n)-4 == a(n)+20 == 0 (mod 3), a(n)-6 == a(n)+22 == 0 (mod 7) and thus a(n) == 97 (mod 210 = 2*3*5*7). (End)
All terms are congruent to 7 (mod 30). - Zak Seidov, May 07 2017
All terms but the first one are congruent to 97 (mod 210). - M. F. Hasler, Jan 18 2022

			n=2: 97, 101, 103, 107, 109, 113 are consecutive primes, while 91, 93, 95 and 115, 117 and 119 are not (cf. 4th comment about the border of composites).

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1000 terms from Zak Seidov)
T. Forbes and Norman Luhn, Prime k-tuplets
R. J. Mathar, Table of Prime Gap Constellations

Cf. A022007.
Cf. A001223, A052162, A052163, A052164, A052165, A052166, A052167, A052168, A047078.
Cf. A350826 (number of n-digit terms).

P:=Filtered([1,3..2*10^7+1],IsPrime);;  I:=[4,2,4,2,4];;
P1:=List([1..Length(P)-1],i->P[i+1]-P[i]);;
A022008:=List(Positions(List([1..Length(P)-Length(I)],i->[P1[i],P1[i+1],P1[i+2],P1[i+3],P1[i+4]]),I),j->P[j]); # Muniru A Asiru, Sep 03 2017

[p: p in PrimesUpTo(2*10^7) | IsPrime(p+4) and IsPrime(p+6) and IsPrime(p+10)and IsPrime(p+12) and IsPrime(p+16)]; // Vincenzo Librandi, Aug 23 2015

for i from 1 to 2*10^5 do if [ithprime(i+1), ithprime(i+2), ithprime(i+3), ithprime(i+4), ithprime(i+5)] = [ithprime(i)+4,ithprime(i)+6,ithprime(i)+10,ithprime(i)+12,ithprime(i)+16] then print(ithprime(i)); fi; od; # Muniru A Asiru, Sep 03 2017

lst = {}; Do[p = Prime[n]; If[PrimeQ[p+4] && PrimeQ[p+6] && PrimeQ[p+10] && PrimeQ[p+12] && PrimeQ[p+16], AppendTo[lst, p]], {n, 1000000}]; lst
Transpose[Select[Partition[Prime[Range[10^6]],6,1],Differences[#]=={4,2,4,2,4}&]][[1]] (* Harvey P. Dale, Mar 15 2015 *)

p=2;q=3;r=5;s=7;t=11;forprime(u=13,1e9,if(u-p==16 && p%3==1, print1(p", "));p=q;q=r;r=s;s=t;t=u) \\ Charles R Greathouse IV, Mar 29 2013

{next_A022008(p, L=Vec(p+1,5), m=210, r=Mod(97,m))=for(i=1,oo, L[i%5+1]+16==(p=nextprime(p+1))&&break; p%m>111 && until(r==p=nextprime((p+8)\/210*210+97),); L[i%5+1]=p); p-16} \\ M. F. Hasler, Jan 18 2022

use ntheory ":all"; say for sieve_prime_cluster(1,1e8, 4,6,10,12,16); # Dana Jacobsen, Sep 30 2015

A078946 Primes p such that p, p+2, p+6, p+12 and p+14 are consecutive primes.

Original entry on oeis.org

17, 227, 1277, 1607, 3527, 3917, 4637, 4787, 27737, 38447, 39227, 44267, 71327, 97367, 99707, 113147, 122027, 122387, 124337, 165707, 183497, 187127, 191447, 197957, 198827, 275447, 290657, 312197, 317957, 347057, 349397, 416387, 418337, 421697, 427067, 443867

Offset: 1

Labos Elemer, Dec 19 2002

nonn

			227 is in the sequence since 227, 229 = 227 + 2, 233 = 227 + 6, 239 = 227 + 12 and 241 = 227 + 14 are consecutive primes.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Eric Weisstein's World of Mathematics, Prime Triplet.

Subsequence of A128468.
Subsequence of A078847. - R. J. Mathar, Feb 10 2013
Cf. A001223, A078866, A078867, A078947-A078971, A022006, A022007.

[p: p in PrimesInInterval(7,1000000) | forall{i: i in [2,6,12,14] | IsPrime(p+i)}]; // Vincenzo Librandi, Apr 19 2015

Rest@ Select[Prime@ Range@ 36000, AllTrue[{2, 6, 12, 14} + #, PrimeQ] &] (* Michael De Vlieger, Apr 18 2015, Version 10 *)
Select[Partition[Prime[Range[36000]],5,1],Differences[#]=={2,4,6,2}&][[All,1]] (* Harvey P. Dale, Jun 14 2022 *)

isok(p) = isprime(p) && (nextprime(p+1)==p+2) && (nextprime(p+3)== p+6) && (nextprime(p+7)==p+12) && (nextprime(p+13)==p+14); \\ Michel Marcus, Dec 10 2013

list(lim) = {my(p1 = 2, p2 = 3, p3 = 5, p4 = 7); forprime(p5 = 11, lim, if(p2 - p1 == 2 && p3 - p2 == 4 && p4 - p3 == 6 && p5 - p4 == 2, print1(p1, ", ")); p1 = p2; p2 = p3; p3 = p4; p4 = p5);} \\ Amiram Eldar, Feb 21 2025

a(n) == 17 (mod 30). - Amiram Eldar, Feb 21 2025

Edited by Dean Hickerson, Dec 20 2002

A086140 Primes p such that three (the maximum number) primes occur between p and p+12.

Original entry on oeis.org

5, 7, 11, 97, 101, 1481, 1867, 3457, 5647, 15727, 16057, 16061, 19417, 19421, 21011, 22271, 43777, 43781, 55331, 79687, 88807, 101107, 144161, 165701, 166841, 195731, 201821, 225341, 247601, 257857, 266677, 268811, 276037, 284737, 326141, 340927

Offset: 1

Labos Elemer, Jul 29 2003

nonn

p+12 must be a prime. - Harvey P. Dale, Jun 11 2015

A086140 is the union of A022006 and A022007. By merging the two b-files I have extended the current b-file up to n=10000 (nearly n=20000 would have been possible). I add a comparison (see Links) between the frequency of prime 5-tuples and an asymptotic approximation, which is unproven but likely to be true, and based on a conjecture first published by Hardy and Littlewood in 1923. Twins, triples and quadruplets are treated as well. - Gerhard Kirchner, Dec 07 2016

			There are two types of prime 5-tuples, and both are represented in this sequence. (11, 13, 17, 19, 23) is a prime 5-tuple of the form (p, p+2, p+6, p+8, p+12), so 11 is in the sequence, and (97, 101, 103, 107, 109) is a prime 5-tuple of the form (p, p+4, p+6, p+10, p+12), so 97 is in the sequence. - _Michael B. Porter_, Dec 19 2016

Harvey P. Dale and Gerhard Kirchner, Table of n, a(n) for n = 1..10000 (first 1000 terms from Harvey P. Dale)
Gerhard Kirchner, Comparison with an assumed asymptotic distribution

Cf. A031930, A046133, A086139, A086136, A022006, A022007, A001359 (twins), A007529 (triples), A007530 (quadruplets).

cp[x_, y_] := Count[Table[PrimeQ[i], {i, x, y}], True] {d=12, k=0}; Do[s=Prime[n]; s1=Prime[n+1]; If[PrimeQ[s+d]&&Equal[cp[s+1, s+d-1], 3], k=k+1; Print[s]], {n, 1, 100000}]
(* Second program: *)
Transpose[Select[Partition[Prime[Range[30000]],5,1],#[[5]]-#[[1]] == 12&]][[1]] (* Harvey P. Dale, Jun 11 2015 *)

A257124 Initial members of prime septuplets.

Original entry on oeis.org

11, 5639, 88799, 165701, 284729, 626609, 855719, 1068701, 1146779, 6560999, 7540439, 8573429, 11900501, 15760091, 17843459, 18504371, 19089599, 21036131, 24001709, 25658441, 39431921, 42981929, 43534019, 45002591, 67816361, 69156539, 74266259, 79208399, 80427029, 84104549, 86818211, 87988709, 93625991, 124066079

Offset: 1

Tim Johannes Ohrtmann, Apr 16 2015

nonn

Tim Johannes Ohrtmann, Table of n, a(n) for n = 1..1990
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: this sequence 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, A257377.
Cf. A343637 (distance from 10^n to the next septuplet).
Cf. A100418.

Disjoint union of A022009 and A022010. - M. F. Hasler, Aug 04 2021

A078866 The quadruples (d1,d2,d3,d4) with elements in {2,4,6} are listed in lexicographic order; for each quadruple, this sequence lists the smallest prime p >= 5 such that the differences between the 5 consecutive primes starting with p are (d1,d2,d3,d4), if such a prime exists.

Original entry on oeis.org

5, 17, 41, 29, 71, 149, 3299, 7, 13, 67, 1597, 19, 43, 12637, 1601, 23, 593, 31, 61, 3313, 157, 47, 601, 151, 251, 3301

Offset: 1

Labos Elemer, Dec 19 2002

The 26 quadruples for which p exists are listed, in decimal form, in A078868.

			The term 12637 corresponds to the quadruple (4,6,6,6): 12637, 12641, 12647, 12653 and 12659 are consecutive primes.

The quadruples are in A078868. The same primes, in increasing order, are in A078867. The sequences of primes corresponding to the 26 difference patterns are in A022006, A022007 and A078946-A078970. Cf. A001223.

Edited by Dean Hickerson, Dec 20 2002

A078867 Sorted version of A078866.

Original entry on oeis.org

5, 7, 13, 17, 19, 23, 29, 31, 41, 43, 47, 61, 67, 71, 149, 151, 157, 251, 593, 601, 1597, 1601, 3299, 3301, 3313, 12637

Offset: 1

Labos Elemer, Dec 19 2002

Each term is the smallest prime p >= 5 such that the differences between the 5 consecutive primes starting with p are (d1,d2,d3,d4), for some quadruple (d1,d2,d3,d4) with elements in {2,4,6}.

			The term 3299 corresponds to the quadruple (2,6,6,6): 3299, 3301, 3307, 3313, 3319 are consecutive primes.

The quadruples are in A078868. The same primes, in lexicographic order of the quadruples, are in A078866. The sequences of primes corresponding to the 26 difference patterns are in A022006, A022007 and A078946-A078970. Cf. A001223.

Edited by Dean Hickerson, Dec 20 2002

cp's OEIS Frontend

A375264 Products of prime 5-tuples (p, p+4, p+6, p+10, p+12) where p = A022007(n).

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A199004 Position of n-th prime sextuplet A022008(n) among prime 5-tuples A022007.

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A144940 Primes of form: sum of prime 5-tuples (A022007).

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A022006 Initial members p of prime 5-tuples (p, p+2, p+6, p+8, p+12).

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A022008 Initial member of prime sextuples (p, p+4, p+6, p+10, p+12, p+16).

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A078946 Primes p such that p, p+2, p+6, p+12 and p+14 are consecutive primes.

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A086140 Primes p such that three (the maximum number) primes occur between p and p+12.

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