A052167 -id:A052167 - cp's OEIS frontend

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

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

A052165 Primes at which the difference pattern X,2,4,2,Y (X and Y >= 6) occurs in A001223.

Original entry on oeis.org

191, 821, 2081, 3251, 9431, 13001, 15641, 18041, 18911, 25301, 31721, 34841, 51341, 62981, 67211, 69491, 72221, 77261, 81041, 82721, 97841, 99131, 109841, 116531, 119291, 122201, 135461, 157271, 171161, 187631, 194861, 201491, 217361

Offset: 1

Labos Elemer, Jan 26 2000

nonn

All terms == 11 (mod 30). - Robert Israel, Nov 30 2015

			191 is here because 191 + 2 = 193, 191 + 4 + 2 = 197, 191 + 2 + 4 + 2 = 199 are primes; the prime preceding 191 is 181; the prime following 199 is 211; and the corresponding differences are 10 and 12. Thus the d-pattern "around 191" is {10,2,4,2,12}.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A001223, A022008, A047078, A052160, A052162, A052163, A052164, A052167, A052168.

Primes:= select(isprime,[2,seq(i,i=3..10^6,2)]):
Gaps:= Primes[2..-1]-Primes[1..-2]:
Primes[select(t -> Gaps[t] = 2 and Gaps[t+1] = 4 and Gaps[t+2] = 2 and Gaps[t-1] >= 6 and Gaps[t+3]>=6, [$2..nops(Gaps)-3])]; # Robert Israel, Nov 30 2015

With[{x = 6, y = 6, s = Partition[#, 6, 1] &@ Prime@ Range[3*10^4]}, Select[s, And[First@ # >= x, Last@ # >= y, Most@ Rest@ # == {2, 4, 2}] &@ Differences@ # &]][[All, 2]] (* Michael De Vlieger, Oct 26 2017 *)

A102332 Initial prime p introducing a prime sextuplet of consecutive primes as follows: {p, p+10, p+18, p+28, p+36, p+46} with the corresponding prime-difference-pattern is {10,8,10,8,10}.

Original entry on oeis.org

37861, 39181, 324763, 692743, 810391, 945331, 1047961, 1429573, 1513573, 1540813, 1799071, 3463573, 3861223, 3979201, 4536121, 4641001, 5154343, 5445403, 5874853, 7851583, 8820793, 8961373, 8976403, 9302113, 9673351, 10323133, 11074033, 11136883, 11899333, 13505983

Offset: 1

Labos Elemer, Jan 06 2005

nonn

A generalization of primes displayed in A022008.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..200 from Harvey P. Dale)

Cf. A001223, A022008, A052162, A052163, A052164, A052165, A052166, A052167, A052168, A047078, A067140, A067141.

tm=TimeUsed[];ta={{0}};Do[g=n;d1=10;d2=8;d3=10;d4=8;d5=10; s1=Prime[n+1]-Prime[n];s2=Prime[n+2]-Prime[n+1]; s3=Prime[n+3]-Prime[n+2];s4=Prime[n+4]-Prime[n+3]; s5=Prime[n+5]-Prime[n+4];If[Equal[s1, d1]&&Equal[s2, d2]&& Equal[s3, d3]&&Equal[s4, d4]&&Equal[s5, d5], Print[{Prime[n], s1, s2, s3, s4, s5}];ta=Append[ta, Prime[n]]], {n, 1, 10000000}] {ta=Delete[ta, 1], {d1, d2}} {g, TimeUsed[]-tm}
Transpose[Select[Partition[Prime[Range[650000]],6,1],Differences[#]=={10,8,10,8,10}&]][[1]] (* Harvey P. Dale, Oct 18 2013 *)

list(lim) = {my(p1 = 2, p2 = 3, p3 = 5, p4 = 7, p5 = 11); forprime(p6 = 13, lim, if(p2 - p1 == 10 && p3 - p2 == 8 && p4 - p3 == 10 && p5 - p4 == 8 && p6 - p5 == 10, print1(p1, ", ")); p1 = p2; p2 = p3; p3 = p4; p4 = p5; p5 = p6);} \\ Amiram Eldar, Feb 18 2025

a(n) == 1 (mod 6). - Amiram Eldar, Feb 18 2025

Definition corrected by Harvey P. Dale, Oct 18 2013

A102333 Initial terms of quartets of consecutive primes as follows: {p, p+16, p+24, p+40}. The corresponding difference-pattern is {16,8,16}.

Original entry on oeis.org

108247, 121507, 166783, 169567, 178207, 216133, 257053, 258763, 272863, 274123, 372613, 383533, 384343, 396157, 413143, 501577, 562477, 577153, 581353, 635293, 721267, 727273, 738937, 769903, 908113, 917713, 932497, 937903, 965467, 980377, 989647, 1008547, 1126537

Offset: 1

Labos Elemer, Jan 06 2005

nonn

A generalization of A052378.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001223, A022008, A052162, A052163, A052164, A052165, A052166, A052167, A052168, A052378, A067140, A067141, A102332.

Transpose[Select[Partition[Prime[Range[78000]],4,1],Differences[#] == {16,8,16}&]][[1]] (* Harvey P. Dale, Mar 18 2012 *)

list(lim) = {my(p1 = 2, p2 = 3, p3 = 5); forprime(p4 = 7, lim, if(p2 - p1 == 16 && p3 - p2 == 8 && p4 - p3 == 16, print1(p1, ", ")); p1 = p2; p2 = p3; p3 = p4);} \\ Amiram Eldar, Feb 18 2025

a(n) == 1 (mod 6). - Amiram Eldar, Feb 18 2025

A102334 Initial terms of quintuplets of consecutive primes as follows: {p, p+16, p+24, p+40, p+48}. The corresponding difference-pattern is {16,8,16,8}.

Original entry on oeis.org

272863, 274123, 372613, 1394893, 1634293, 2380423, 3846373, 5298523, 5358013, 5797903, 6741913, 7554823, 7647643, 7716103, 7738153, 8241463, 8358283, 9710473, 9859783, 12454333, 12510193, 12796423, 13710133, 14477893, 15162493, 15186583, 15263503, 15603853, 16438243, 16771933, 17913283, 18957973, 19373623

Offset: 1

Labos Elemer, Jan 06 2005

nonn

A generalization of A022007.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001223, A022007, A052162, A052163, A052164, A052165, A052166, A052167, A052168, A052378, A067140, A067141, A102332, A102333.

Select[Partition[Prime[Range[1233300]], 5, 1], Differences[#] == {16, 8, 16, 8} &][[;;, 1]] (* Amiram Eldar, Feb 18 2025 *)

list(lim) = {my(p1 = 2, p2 = 3, p3 = 5, p4 = 7); forprime(p5 = 11, lim, if(p2 - p1 == 16 && p3 - p2 == 8 && p4 - p3 == 16 && p5 - p4 == 8, print1(p1, ", ")); p1 = p2; p2 = p3; p3 = p4; p4 = p5);} \\ Amiram Eldar, Feb 18 2025

a(n) == 13 (mod 30). - Amiram Eldar, Feb 18 2025

Missing terms a(1)-a(11) inserted by Amiram Eldar, Feb 18 2025

A102335 Initial terms of sextuplets of consecutive primes as follows: {p, p+16, p+24, p+40, p+48, p+64}. The corresponding difference-pattern is {16,8,16,8,16}.

Original entry on oeis.org

12454333, 21228553, 25131193, 38589673, 41426353, 46254253, 56564623, 60498133, 61151863, 96691213, 158497153, 169760713, 182960473, 201513133, 226086283, 236031463, 253806913, 290686483, 305472373, 344550643, 369110983, 380973253, 421335883, 445537333, 461955763

Offset: 1

Labos Elemer, Jan 06 2005

nonn

A generalization of A022008. The generalized pattern of consecutive prime-differences is {6a+4, 6b+2, 6c+4, 6d+2, 6e+4} with a = c = e = 2, b = d = 1.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001223, A022008, A052162, A052163, A052164, A052165, A052166, A052167, A052168, A052378, A067140, A067141, A102332, A102333, A102334.

Transpose[Select[Partition[Prime[Range[20000000]],6,1],Differences[#] == {16,8,16,8,16}&]][[1]] (* Harvey P. Dale, Nov 08 2011 *)

list(lim) = {my(p1 = 2, p2 = 3, p3 = 5, p4 = 7, p5 = 11); forprime(p6 = 13, lim, if(p2 - p1 == 16 && p3 - p2 == 8 && p4 - p3 == 16 && p5 - p4 == 8 && p6 - p5 == 16, print1(p1, ", ")); p1 = p2; p2 = p3; p3 = p4; p4 = p5; p5 = p6);} \\ Amiram Eldar, Feb 18 2025

a(n) == 73 (mod 210). - Amiram Eldar, Feb 18 2025

A102331 Initial members of quintuplets (p, p+4, p+12, p+16, p+24) of consecutive primes with the corresponding difference pattern is {4,8,4,8}.

Original entry on oeis.org

13147, 14407, 114757, 132607, 231547, 353317, 459607, 476587, 568987, 601747, 652357, 724627, 794137, 861547, 904777, 1010407, 1094437, 1140847, 1147567, 1170007, 1270417, 1424557, 1441327, 1477027, 1604497, 1646287, 1673377, 2043397, 2078707, 2126767, 2130367

Offset: 1

Labos Elemer, Jan 07 2005

nonn

Generalization of A022007. These primes are congruent to 7 modulo 10, so the realization of longer prime-difference pattern = {4,8,4,8,4} is not already possible because the sum = 4+8+4+8+4 = 28. Consequently, 10k+7+28 = 10m+5 cannot be a prime. Thus analogous generalization of A022008 is possible only with restrictions. See also Comment in A102335.

			The prime 13147 is followed by the primes {13151, 13159, 13163, 13171}. Observe that these patterns start and end with primes of the form 10k+7 and 10m+1, respectively.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001223, A022007, A022008, A052162, A052163, A052164, A052165, A052166, A052167, A052168, A102335.

Select[Partition[Prime[Range[158000]], 5, 1], Differences[#] == {4, 8, 4, 8} &][[;;, 1]] (* Amiram Eldar, Feb 18 2025 *)

list(lim) = {my(p1 = 2, p2 = 3, p3 = 5, p4 = 7); forprime(p5 = 11, lim, if(p2 - p1 == 4 && p3 - p2 == 8 && p4 - p3 == 4 && p5 - p4 == 8, print1(p1, ", ")); p1 = p2; p2 = p3; p3 = p4; p4 = p5);} \\ Amiram Eldar, Feb 18 2025

a(n) == 7 (mod 30). - Amiram Eldar, Feb 18 2025

A102336 Initial members of quintuplets (p, p+4, p+12, p+28, p+60) of consecutive primes with the corresponding difference pattern is {4,8,16,32}.

Original entry on oeis.org

1197739, 2496409, 2692549, 2962489, 3195679, 5723479, 6824899, 7706059, 8056039, 8337319, 10132609, 10583269, 11739589, 12167509, 12674659, 13007959, 13699459, 14148049, 14252929, 14702839, 15726019, 16694539, 17115949, 17282299, 17350159, 17584729, 18065389, 18097609

Offset: 1

Labos Elemer, Jan 07 2005

nonn

Generalization of A022007. These primes are congruent to 9 modulo 10, while terminal entry of 5-tuple has the form 10s+9.

			1197739 is a prime, followed by (1197743, 1197751, 1197767, 1197799) with consecutive prime difference pattern: {4,8,16,32}.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001223, A022007, A022008, A052162, A052163, A052164, A052165, A052166, A052167, A052168, A102331, A102332, A102333, A102334, A102335.

Select[Partition[Prime[Range[10^6]], 5, 1], Differences[#] == 2^Range[2, 5] &][[;;, 1]] (* Amiram Eldar, Feb 18 2025 *)

list(lim) = {my(p1 = 2, p2 = 3, p3 = 5, p4 = 7); forprime(p5 = 11, lim, if(p2 - p1 == 4 && p3 - p2 == 8 && p4 - p3 == 16 && p5 - p4 == 32, print1(p1, ", ")); p1 = p2; p2 = p3; p3 = p4; p4 = p5);} \\ Amiram Eldar, Feb 18 2025

a(n) == 19 (mod 30). - Amiram Eldar, Feb 18 2025

A102337 Initial members of sextuplets (p, p+4, p+12, p+28, p+60, p+124) of consecutive primes with the corresponding difference pattern is {4,8,16,32,64}.

Original entry on oeis.org

166392559, 337149859, 1356705139, 1455488059, 1879518709, 2339605519, 2410687039, 2811378079, 3191346019, 3250560139, 3442915309, 3573582079, 4873308619, 4875167959, 5362448719, 5524743379, 5580251359, 5716641649, 5783545759, 5977816549, 6019275469, 6076905349

Offset: 1

Labos Elemer, Jan 07 2005

nonn

Generalization of A022008 because the relevant prime-difference pattern has the following form: (4+6a,2+6b,4+6c,2+6d,4+6e), a = 0, b = 1, c = 2, d = 5, e = 10. The primes are congruent to 9 modulo 10, while terminal entries of the quintuplets have the form 10s+3.

			1455488059 is a prime, followed by consecutive prime difference pattern: {4,8,16,32,64}. The terminal prime is 1455488183.

Amiram Eldar, Table of n, a(n) for n = 1..2500

Cf. A001223, A022008, A022008, A052162, A052163, A052164, A052165, A052166, A052167, A052168, A102331, A102332, A102333, A102334, A102335, A102336.

Select[Partition[Prime[Range[3*10^7]], 6, 1], Differences[#] == 2^Range[2, 6] &][[;;, 1]] (* Amiram Eldar, Feb 18 2025 *)

list(lim) = {my(p1 = 2, p2 = 3, p3 = 5, p4 = 7, p5 = 11); forprime(p6 = 13, lim, if(p2 - p1 == 4 && p3 - p2 == 8 && p4 - p3 == 16 && p5 - p4 == 32 && p6 - p5 == 64, print1(p1, ", ")); p1 = p2; p2 = p3; p3 = p4; p4 = p5; p5 = p6);} \\ Amiram Eldar, Feb 18 2025

a(n) == 19 (mod 30). - Amiram Eldar, Feb 18 2025

a(5)-a(18) from Donovan Johnson, Apr 17 2010

a(19)-a(22) from Amiram Eldar, Feb 18 2025

cp's OEIS Frontend

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

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A052165 Primes at which the difference pattern X,2,4,2,Y (X and Y >= 6) occurs in A001223.

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A102332 Initial prime p introducing a prime sextuplet of consecutive primes as follows: {p, p+10, p+18, p+28, p+36, p+46} with the corresponding prime-difference-pattern is {10,8,10,8,10}.

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A102333 Initial terms of quartets of consecutive primes as follows: {p, p+16, p+24, p+40}. The corresponding difference-pattern is {16,8,16}.

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A102334 Initial terms of quintuplets of consecutive primes as follows: {p, p+16, p+24, p+40, p+48}. The corresponding difference-pattern is {16,8,16,8}.

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A102335 Initial terms of sextuplets of consecutive primes as follows: {p, p+16, p+24, p+40, p+48, p+64}. The corresponding difference-pattern is {16,8,16,8,16}.

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A102331 Initial members of quintuplets (p, p+4, p+12, p+16, p+24) of consecutive primes with the corresponding difference pattern is {4,8,4,8}.

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