A078866 -id:A078866 - cp's OEIS frontend

A078867 Sorted version of A078866.

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

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

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

A078868 Decimal concatenations of the quadruples (d1,d2,d3,d4) with elements in {2,4,6} for which there exists a prime p >= 5 such that the differences between the 5 consecutive primes starting with p are (d1,d2,d3,d4).

Original entry on oeis.org

2424, 2462, 2466, 2642, 2646, 2664, 2666, 4242, 4246, 4264, 4624, 4626, 4662, 4666, 6246, 6264, 6266, 6424, 6426, 6462, 6466, 6626, 6642, 6646, 6662, 6664

Offset: 1

Labos Elemer, Dec 19 2002

			4624 corresponds to the quadruple (4,6,2,4). It is in the sequence because the 5 consecutive primes 1597, 1601, 1607, 1609 and 1613 have differences (4,6,2,4).

The least primes corresponding to the quadruples are in A078866. The same primes, in increasing order, are in A078867. The sequences of primes corresponding to the 26 difference patterns are in A022006 (for 2424), A022007 (for 4242) and A078946-A078970. The similarly defined quintuples are in A078870. Cf. A001223.

With[{k = 4}, FromDigits /@ Select[Tuples[Range[2, 6, 2], k], Function[m, Count[Range[k, 10^k], n_ /; Times @@ Boole@ Map[PrimeQ, Prime@ n + Accumulate@ m] == 1] > 0]]] (* Michael De Vlieger, Mar 25 2017 *) (* or *)
FromDigits /@ Union@ Select[ Partition[ Differences@ Prime@ Range[3, 2000], 4, 1], Max@ # <= 6 &] (* Giovanni Resta, Mar 25 2017 *)

Edited by Dean Hickerson, Dec 20 2002

A078872 The quintuples (d1,d2,d3,d4,d5) with elements in {2,4,6} are listed in lexicographic order; for each quintuple, this sequence lists the smallest prime p >= 7 such that the differences between the 6 consecutive primes starting with p are (d1,d2,d3,d4,d5), if such a prime exists.

Original entry on oeis.org

11, 17, 41, 29, 59, 5849, 6959, 599, 149, 3299, 7, 13, 37, 67, 1597, 19, 4639, 43, 17467, 1601, 23, 2333, 593, 6353, 1861, 31, 61, 90001, 32353, 157, 14731, 47, 587, 2671, 3307, 151, 251, 3301

Offset: 1

Labos Elemer, Dec 20 2002

Comment from N. J. A. Sloane, Dec 04 2015: (Start)
The definition of A078872 is fairly subtle.
Step 1: The 3^5 = 243 quintuples (d1,d2,d3,d4,d5) with elements in {2,4,6} are listed in lexicographic order.
Step 2: Study each quintuple in turn.  Look for  the smallest prime p >= 7 such that the differences between the 6 consecutive primes starting with p are (d1,d2,d3,d4,d5).  If there is no such prime move on to next quintuple. If there is at least one such prime, take the smallest one, add it to the sequence, and move on to the next quintuple.
Each quintuple is considered just once, so there are at most 243 terms (in fact there are only 38).
(End)
The 38 quintuples for which p exists are listed, in decimal form, in A078870.

			The term 67 corresponds to the quintuple (4,2,6,4,6): 67, 71, 73, 79, 83 and 89 are consecutive primes.

The quintuples are in A078870. The same primes, in increasing order, are in A078873. The analogous sequences for quadruples and 6-tuples are in A078866 and A078874. Cf. A001223.

Edited by Dean Hickerson, Dec 21 2002

A078874 The 6-tuples (d1,d2,d3,d4,d5,d6) with elements in {2,4,6} are listed in lexicographic order; for each 6-tuple, this sequence lists the smallest prime p >= 7 such that the differences between the 7 consecutive primes starting with p are (d1,d2,d3,d4,d5,d6), if such a prime exists.

Original entry on oeis.org

11, 17, 4637, 41, 5639, 29, 59, 130631, 78779, 603899, 149, 3299, 13, 37, 1597, 19, 5839, 135589, 71329, 43, 302563, 17467, 1601, 23, 53, 5843, 326993, 593, 135593, 71333, 44257, 31, 61, 678631, 32353, 435553, 6268957, 351031, 47, 41597, 587, 19457, 2671, 246907, 151, 251, 179801, 3301

Offset: 1

Labos Elemer, Dec 20 2002

The 48 6-tuples for which p exists are listed, in decimal form, in A078871.

			The term 151 corresponds to the 6-tuple (6,6,4,6,6,2): 151, 157, 163, 167, 173, 179, 181 are consecutive primes.

The 6-tuples are in A078871. The same primes, in increasing order, are in A078875. The analogous sequences for quadruples and quintuples are in A078866 and A078872. Cf. A001223.

Edited by Dean Hickerson, Dec 21 2002

A078947 Primes p such that the differences between the 5 consecutive primes starting with p are (2,4,6,6).

Original entry on oeis.org

41, 641, 1091, 4001, 9461, 26681, 26711, 44531, 79811, 103991, 110921, 112571, 172421, 223241, 276821, 289841, 290021, 317771, 373181, 381371, 434921, 450881, 493121, 602081, 678761, 788351, 834131, 907211, 974861, 1076501, 1081121, 1097891, 1200371, 1409531, 1426151

Offset: 1

Labos Elemer, Dec 19 2002

nonn

Equivalently, primes p such that p, p+2, p+6, p+12 and p+18 are consecutive primes.

			641 is in the sequence since 641, 643 = 641 + 2, 647 = 641 + 6, 653 = 641 + 12 and 659 = 641 + 18 are consecutive primes.

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

Subsequence of A078847.

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

Select[Partition[Prime[Range[50000]], 5, 1], Differences[#] == {2, 4, 6, 6} &][[;;, 1]] (* Amiram Eldar, Feb 21 2025 *)

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 == 6, print1(p1, ", ")); p1 = p2; p2 = p3; p3 = p4; p4 = p5);} \\ Amiram Eldar, Feb 21 2025

a(n) == 11 (mod 30). - Amiram Eldar, Feb 21 2025

Edited by Dean Hickerson, Dec 20 2002

A078948 Primes p such that the differences between the 5 consecutive primes starting with p are (2,6,4,2).

Original entry on oeis.org

29, 59, 269, 1289, 2129, 2789, 5639, 8999, 13679, 14549, 18119, 36779, 62129, 75989, 80669, 83219, 88799, 93479, 113159, 115769, 124769, 132749, 150209, 160079, 163979, 203309, 207509, 223829, 228509, 278489, 282089, 284729, 298679, 312929, 313979, 323369, 337859

Offset: 1

Labos Elemer, Dec 19 2002

nonn

Equivalently, primes p such that p, p+2, p+8, p+12 and p+14 are consecutive primes.

All terms are congruent to 29 (mod 30). - Muniru A Asiru, Sep 04 2017

			59 is in the sequence since 59, 61 = 59 + 2, 67 = 59 + 8, 71 = 59 + 12 and 73 = 59 + 14 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 A078848. - R. J. Mathar, Feb 10 2013

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

K:=26*10^7+1;; # to get all terms <= K.
P:=Filtered([1,3..K],IsPrime);;  I:=[2,6,4,2];;
P1:=List([1..Length(P)-1],i->P[i+1]-P[i]);;
Q:=List(Positions(List([1..Length(P)-Length(I)],i->[P1[i],P1[i+1],P1[i+2],P1[i+3]]),I),i->P[i]); # Muniru A Asiru, Sep 04 2017

for i from 1 to 10^5 do if [ithprime(i+1),ithprime(i+2),ithprime(i+3),ithprime(i+4)] = [ithprime(i)+2,ithprime(i)+8,ithprime(i)+12,ithprime(i)+14] then print(ithprime(i)); fi; od;  # Muniru A Asiru, Sep 04 2017

Select[Partition[Prime[Range[26000]],5,1],Differences[#]=={2,6,4,2}&][[;;,1]] (* Harvey P. Dale, Dec 10 2024 *)

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

Edited by Dean Hickerson, Dec 20 2002

A078949 Primes p such that the differences between the 5 consecutive primes starting with p are (2,6,4,6).

Original entry on oeis.org

71, 431, 2339, 2381, 5849, 6959, 27791, 32561, 41609, 45119, 46439, 48479, 51419, 54401, 63599, 78779, 81551, 106859, 115319, 130631, 138569, 143501, 153269, 166601, 183569, 196169, 204359, 229751, 246929, 266081, 279119, 321311, 326999, 350729, 357659, 362741

Offset: 1

Labos Elemer, Dec 19 2002

nonn

Equivalently, primes p such that p, p+2, p+8, p+12 and p+18 are consecutive primes.

			71 is in the sequence since 71, 73 = 71 + 2, 79 = 71 + 8, 83 = 71 + 12 and 89 = 71 + 18 are consecutive primes.

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

Subsequence of A078848. - R. J. Mathar, Feb 10 2013

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

Select[Partition[Prime[Range[50000]], 5, 1], Differences[#] == {2, 6, 4, 6} &][[;;, 1]] (* Amiram Eldar, Feb 21 2025 *)

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

From Amiram Eldar, Feb 21 2025: (Start)
a(n) == 5 (mod 6).
a(n) == 11 or 29 (mod 30). (End)

Edited by Dean Hickerson, Dec 20 2002

A078950 Primes p such that the differences between the 5 consecutive primes starting with p are (2,6,6,4).

Original entry on oeis.org

149, 599, 27059, 31319, 42179, 65699, 75209, 85829, 87539, 92219, 135599, 170759, 205949, 221069, 249419, 274829, 278609, 280589, 287849, 302579, 307259, 308309, 350429, 355499, 398339, 406499, 416399, 422549, 541529, 566549, 573479, 585839, 603899, 609599, 637709

Offset: 1

Labos Elemer, Dec 19 2002

nonn

Equivalently, primes p such that p, p+2, p+8, p+14 and p+18 are consecutive primes.

			149 is in the sequence since 149, 151 = 149 + 2, 157 = 149 + 8, 163 = 149 + 14 and 167 = 149 + 18 are consecutive primes.

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

Subsequence of A078849. - R. J. Mathar, May 06 2017

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

Select[Partition[Prime[Range[50000]], 5, 1], Differences[#] == {2, 6, 6, 4} &][[;;, 1]] (* Amiram Eldar, Feb 21 2025 *)

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

a(n) == 29 (mod 30). - Amiram Eldar, Feb 21 2025

Edited by Dean Hickerson, Dec 20 2002

cp's OEIS Frontend

A078867 Sorted version of A078866.

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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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A078969 Primes p such that the differences between the 5 consecutive primes starting with p are (6,6,6,4).

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A078868 Decimal concatenations of the quadruples (d1,d2,d3,d4) with elements in {2,4,6} for which there exists a prime p >= 5 such that the differences between the 5 consecutive primes starting with p are (d1,d2,d3,d4).

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A078872 The quintuples (d1,d2,d3,d4,d5) with elements in {2,4,6} are listed in lexicographic order; for each quintuple, this sequence lists the smallest prime p >= 7 such that the differences between the 6 consecutive primes starting with p are (d1,d2,d3,d4,d5), if such a prime exists.

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A078874 The 6-tuples (d1,d2,d3,d4,d5,d6) with elements in {2,4,6} are listed in lexicographic order; for each 6-tuple, this sequence lists the smallest prime p >= 7 such that the differences between the 7 consecutive primes starting with p are (d1,d2,d3,d4,d5,d6), if such a prime exists.

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A078947 Primes p such that the differences between the 5 consecutive primes starting with p are (2,4,6,6).

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Mathematica

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A078948 Primes p such that the differences between the 5 consecutive primes starting with p are (2,6,4,2).

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