A052378 -id:A052378 - cp's OEIS frontend

A078849 Initial term in sequence of four consecutive primes separated by 3 consecutive differences each <=6 (i.e., when d=2,4 or 6) and forming d-pattern=[2, 6,6]; short d-string notation of pattern = [266].

149, 599, 3299, 4649, 5099, 6359, 11489, 12539, 16979, 19469, 27059, 30089, 31319, 34259, 42179, 53609, 58229, 63689, 65699, 71339, 75209, 77549, 78569, 80909, 81929, 85829, 87509, 87539, 89519, 92219, 101279, 105359, 112289, 116099, 116789

Offset: 1

Labos Elemer, Dec 11 2002

nonn

Subsequence of A049437. - R. J. Mathar, Feb 10 2013

			149, 149+2=151, 149+2+6=157, 149+2+6+6=163 are consecutive primes.

R. J. Mathar, Table of n, a(n) for n = 1..1000

Cf. analogous prime quadruple sequences with various possible {2, 4, 6}-difference-patterns in brackets: A007530[242], A078847[246], A078848[264], A078849[266], A052378[424], A078850[426], A078851[462], A078852[466], A078853[624], A078854[626], A078855[642], A078856[646], A078857[662], A078858[664], A033451[666].

d = {2, 6, 6}; First /@ Select[Partition[Prime@ Range@ 12000, Length@ d + 1, 1], Differences@ # == d &] (* Michael De Vlieger, May 02 2016 *)
Select[Partition[Prime[Range[12000]],4,1],Differences[#]=={2,6,6}&][[All,1]] (* Harvey P. Dale, Dec 29 2017 *)

Primes p = p(i) such that p(i+1)=p+2, p(i+2)=p+2+6, p(i+3)=p+2+6+6.

Listed terms verified by Ray Chandler, Apr 20 2009

A078853 Initial term in sequence of four consecutive primes separated by 3 consecutive differences each <=6 (i.e., when d = 2, 4 or 6) and forming d-pattern=[6,2,4]; short d-string notation of pattern = [624].

Original entry on oeis.org

1601, 3911, 5471, 8081, 12101, 12911, 13751, 14621, 17021, 32051, 38321, 40841, 43391, 58901, 65831, 67421, 67751, 68891, 69821, 72161, 80141, 89591, 90011, 90191, 97571, 100511, 102191, 111821, 112241, 122021, 125921, 129281, 129581

Offset: 1

Labos Elemer, Dec 11 2002

nonn

All terms are == 11 (mod 30). Is 180 the minimal first difference? - Zak Seidov, Jun 27 2015

Subsequence of A049438. - R. J. Mathar, May 06 2017

			p=1601, 1601+6=1607, 1601+6+2=1609, 1601+6+2+4=1613 are consecutive primes.

R. J. Mathar, Table of n, a(n) for n = 1..1000

Cf. analogous prime quadruple sequences with various possible {2, 4, 6}-difference-patterns in brackets: A007530[242], A078847[246], A078848[264], A078849[266], A052378[424], A078850[426], A078851[462], A078852[466], this sequence[624], A078854[626], A078855[642], A078856[646], A078857[662], A078858[664], A033451[666].

Transpose[Select[Partition[Prime[Range[13000]], 4, 1], Differences[#]=={6, 2, 4} &]][[1]] (* Vincenzo Librandi, Jun 27 2015 *)

Primes p = p(i) such that p(i+1)=p+6, p(i+2)=p+6+2, p(i+3)=p+6+2+4.

Listed terms verified by Ray Chandler, Apr 20 2009

A078850 Initial term in sequence of four consecutive primes separated by 3 consecutive differences each <=6 (i.e., when d=2,4 or 6) and forming d-pattern=[4,2,6]; short d-string notation of pattern = [426].

Original entry on oeis.org

67, 1447, 2377, 2707, 5437, 5737, 7207, 9337, 11827, 12037, 19207, 21487, 21517, 23197, 26107, 26947, 28657, 31147, 31177, 35797, 37357, 37567, 42697, 50587, 52177, 65167, 67927, 69997, 71707, 74197, 79147, 81547, 103087, 103387, 106657

Offset: 1

Labos Elemer, Dec 11 2002

nonn

Subsequence of A022005. - R. J. Mathar, May 06 2017

			p=67,67+4=71,67+4+2=73,67+4+2+6=79 are consecutive primes.

R. J. Mathar, Table of n, a(n) for n = 1..1000

Cf. analogous prime quadruple sequences with various possible {2, 4, 6}-difference-patterns in brackets: A007530[242], A078847[246], A078848[264], A078849[266], A052378[424], A078850[426], A078851[462], A078852[466], A078853[624], A078854[626], A078855[642], A078856[646], A078857[662], A078858[664], A033451[666].

d = {4, 2, 6}; First /@ Select[Partition[Prime@ Range@ 12000, Length@ d + 1, 1], Differences@ # == d &] (* Michael De Vlieger, May 02 2016 *)

Primes p = p(i) such that p(i+1)=p+4, p(i+2)=p+4+2, p(i+3)=p+4+2+6.

Listed terms verified by Ray Chandler, Apr 20 2009

A052376 Primes followed by a [10,2,10] prime difference pattern of A001223.

Original entry on oeis.org

409, 1039, 2017, 2719, 3571, 4219, 4231, 4261, 4327, 6079, 6121, 6679, 6781, 8209, 11047, 11149, 11959, 12241, 15277, 19531, 19687, 21577, 21589, 26881, 27529, 28087, 28297, 29389, 30829, 30859, 31069, 32401, 42061, 45307, 47797, 48109

Offset: 1

Labos Elemer, Mar 22 2000

nonn

Subsequence of lesser terms of 10-twins (A031928).
Start primes of quadruples consisting of two consecutive 10-twins of prime which are in minimal distance [minD = A052380(10/2) = 12].
First term of this sequence is 409 = A052381(5).

			p=1039 begins [1039,1049,1051,1061] prime quadruple with the appropriate difference pattern: [10,2,10]=[d,D-d,d], so d=10, D=12.

Robert G. Wilson v, Table of n, a(n) for n = 1..1780

Cf. A031928, A053323, A052380, A052381, A052377, A052378.

{p, q, r, s} = {2, 3, 5, 7}; lst = {}; While[p < 50000, If[ Differences[{p, q, r, s}] == {10, 2, 10}, AppendTo[lst, p]]; {p, q, r, s} = {q, r, s, NextPrime@ s}]; lst (* Robert G. Wilson v, Jul 15 2015 *)

a(n)=p, a prime which begins a [p, p+d, p+D, p+D+d]=[p, p+10, p+12, p+22] prime quadruple.

a(n) = A259025(n)-11. - Robert G. Wilson v, Jul 15 2015

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

A052377 Primes followed by an [8,4,8]=[d,D-d,d] prime difference pattern of A001223.

Original entry on oeis.org

389, 479, 1559, 3209, 8669, 12269, 12401, 13151, 14411, 14759, 21851, 28859, 31469, 33191, 36551, 39659, 40751, 50321, 54311, 64601, 70229, 77339, 79601, 87671, 99551, 102539, 110261, 114749, 114761, 118661, 129449, 132611, 136511

Offset: 1

Labos Elemer, Mar 22 2000

nonn

A subsequence of A031926. [Corrected by Sean A. Irvine, Nov 07 2021]
a(n)=p, the initial prime of two consecutive 8-twins of primes as follows: [p,p+8] and [p+12,p+12+8], d=8, while the distance of the two 8-twins is 12 (minimal; see A052380(4/2)=12).
Analogous sequences are A047948 for d=2, A052378 for d=4, A052376 for d=10 and A052188-A052199 for d=6k, so that in the [d,D-d,d] difference patterns which follows a(n) the D-d is minimal(=0,2,4; here it is 4).

			p=1559 begins the [1559,1567,1571,1579] prime quadruple consisting of two 8-twins [1559,1567] and[1571,1579] which are in minimal distance, min{D}=1571-1559=12=A052380(8/2).

Cf. A031926, A053325, A052380, A052376, A052378, A052188-A052190.

a(n) is the initial term of a [p, p+8, p+12, p+12+8] quadruple of consecutive primes.

A078952 Primes p such that the differences between the 5 consecutive primes starting with p are (4,2,4,6).

Original entry on oeis.org

13, 37, 223, 1087, 1423, 1483, 2683, 4783, 20743, 27733, 29017, 33343, 33613, 35527, 42457, 44263, 45817, 55813, 93487, 108877, 110917, 113143, 118897, 151237, 165703, 187123, 198823, 203653, 205417, 221713, 234187, 234457, 258607, 276817, 284227, 289837, 308923

Offset: 1

Labos Elemer, Dec 19 2002

nonn

Equivalently, primes p such that p, p+4, p+6, p+10 and p+16 are consecutive primes.

All terms = {7, 13} mod 30. - Muniru A Asiru, Aug 21 2017

			37 is in the sequence since 37, 41 = 37 + 4, 43 = 37 + 6, 47 = 37 + 10 and 53 = 37 + 16 are consecutive primes.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from R. J. Mathar)
R. J. Mathar, Table of Prime Gap Constellations.

Subsequence of A052378. - R. J. Mathar, Feb 11 2013

Cf. A001223, A022006, A022007, A078866, A078867, A078946-A078971.

K:=2*10^7+1;; # to get all terms <= K.
P:=Filtered([1,3..K],IsPrime);;  I:=[4,2,4,6];;
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]]);;
P3:=List(Positions(P2,I),i->P[i]); # Muniru A Asiru, Aug 21 2017

for i from 1 to 10^7 do if ithprime(i+1)=ithprime(i)+4 and ithprime(i+2)=ithprime(i)+6 and ithprime(i+3)=ithprime(i)+10 and ithprime(i+4)=ithprime(i)+16 then print(ithprime(i)); fi; od; # Muniru A Asiru, Aug 21 2017

With[{s = Differences@ Prime@ Range[10^5]}, Prime[SequencePosition[s, {4, 2, 4, 6}][[All, 1]]]] (* Michael De Vlieger, Aug 21 2017 *)

lista(nn) = forprime(p=3, nn, if(nextprime(p+1)==p+4 && nextprime(p+5)==p+6 && nextprime(p+7)==p+10 && nextprime(p+11)==p+16, print1(p, ", "))); \\ Altug Alkan, Aug 21 2017

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

Edited by Dean Hickerson, Dec 20 2002

A384298 Primes p such that p + 4, p + 12 and p + 16 are also primes.

Original entry on oeis.org

7, 67, 97, 487, 757, 1567, 1597, 2377, 3907, 7687, 8677, 12097, 12907, 13147, 14407, 14767, 15667, 16057, 19417, 21487, 31177, 38317, 43777, 52567, 57637, 58897, 65167, 65827, 67477, 67927, 74857, 81547, 90007, 90187, 93967, 94777, 95467, 95617, 102547, 111427, 112237, 114757, 123817, 129277

Offset: 1

Alexander Yutkin, May 25 2025

nonn

Initial members of prime quartets that correspond to the difference pattern [4, 8, 4].

			p=97: 97+4=101, 97+12=109, 97+16=113 —> prime quartet: (97, 101, 109, 113).

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

Cf. A000040, A001223.

Cf. A136162 [2, 4, 2], A052378 [4, 2, 4], A382810 [6, 4, 6].

q:= n-> andmap(i-> isprime(n+4*i), [0,1,3,4]):
select(q, [7+30*i$i=0..4309])[];  # Alois P. Heinz, May 29 2025

Select[Prime[Range[12099]],AllTrue[#+{4,12,16},PrimeQ]&] (* James C. McMahon, May 29 2025 *)

a(n) == 7 (mod 30).

cp's OEIS Frontend

A078849 Initial term in sequence of four consecutive primes separated by 3 consecutive differences each <=6 (i.e., when d=2,4 or 6) and forming d-pattern=[2, 6,6]; short d-string notation of pattern = [266].

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A078853 Initial term in sequence of four consecutive primes separated by 3 consecutive differences each <=6 (i.e., when d = 2, 4 or 6) and forming d-pattern=[6,2,4]; short d-string notation of pattern = [624].

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A078850 Initial term in sequence of four consecutive primes separated by 3 consecutive differences each <=6 (i.e., when d=2,4 or 6) and forming d-pattern=[4,2,6]; short d-string notation of pattern = [426].

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A052376 Primes followed by a [10,2,10] prime difference pattern of A001223.

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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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