A078855 -id:A078855 - cp's OEIS frontend

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].

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

A079016 Suppose p and q = p+12 are primes. Define the difference pattern of (p,q) to be the successive differences of the primes in the range p to q. There are 14 possible difference patterns, namely [12], [2,10], [4,8], [6,6], [8,4], [10,2], [2,4,6], [2,6,4], [4,2,6], [4,6,2], [6,2,4], [6,4,2], [2,4,2,4] and [4,2,4,2]. Sequence gives smallest value of p for each difference pattern, sorted by magnitude.

Original entry on oeis.org

5, 7, 17, 19, 29, 31, 47, 67, 89, 137, 139, 199, 397, 1601

Offset: 1

Labos Elemer, Jan 24 2003

			p=1601, q=1613 has difference pattern [6,2,4] and {1601,1607,1609,1613} is the corresponding consecutive prime 4-tuple.

A022006(1)=5, A022007(1)=7, A078847(1)=17, A078851(1)=19, A078848(1)=29, A078855(1)=31, A047948(1)=47, A078850(1)=67, A031930(1)=A000230(6)=199, A046137(1)=7, A078853(1)=1601.

Cf. A079017-A079024.

Function[s, Function[t, Union@ Flatten@ Map[s[[First@ Position[t, #]]] &, {{12}, {2, 10}, {4, 8}, {6, 6}, {8, 4}, {10, 2}, {2, 4, 6}, {2, 6, 4}, {4, 2, 6}, {4, 6, 2}, {6, 2, 4}, {6, 4, 2}, {2, 4, 2, 4}, {4, 2, 4, 2}}]]@ Map[Differences@ Select[Range[#, # + 12], PrimeQ] &, s]]@ Select[Prime@ Range[10^3], PrimeQ[# + 12] &] (* Michael De Vlieger, Feb 25 2017 *)

A078961 Primes p such that the differences between the 5 consecutive primes starting with p are (6,4,2,4).

Original entry on oeis.org

31, 1291, 1861, 1987, 2677, 4507, 5641, 7867, 13681, 17377, 24097, 35521, 42451, 44257, 55807, 80671, 88651, 88801, 93481, 110557, 113011, 113161, 118891, 134581, 155371, 163981, 198817, 221707, 234181, 266671, 269377, 284731, 290611, 313981, 331537, 332461, 344161

Offset: 1

Labos Elemer, Dec 19 2002

nonn

Equivalently, primes p such that p, p+6, p+10, p+12 and p+16 are consecutive primes.

			31 is a term since 31, 37 = 31 + 6, 41 = 31 + 10, 43 = 31 + 12 and 47 = 31 + 16 are consecutive primes.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)

Subsequence of A078855. - R. J. Mathar, May 06 2017

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

Transpose[Select[Partition[Prime[Range[26000]],5,1],Differences[#]=={6,4,2,4}&]][[1]] (* Harvey P. Dale, Aug 26 2014 *)

list(lim) = {my(p1 = 2, p2 = 3, p3 = 5, p4 = 7); forprime(p5 = 11, lim, if(p2 - p1 == 6 && p3 - p2 == 4 && p4 - p3 == 2 && p5 - p4 == 4, print1(p1, ", ")); p1 = p2; p2 = p3; p3 = p4; p4 = p5);} \\ Amiram Eldar, Feb 22 2025

From Amiram Eldar, Feb 22 2025: (Start)
a(n) == 1 (mod 6).
a(n) == 1 or 7 (mod 30). (End)

Edited by Dean Hickerson, Dec 20 2002

A078962 Primes p such that the differences between the 5 consecutive primes starting with p are (6,4,2,6).

Original entry on oeis.org

61, 2371, 5431, 11821, 21481, 37561, 50581, 69991, 124291, 126481, 139291, 223831, 230761, 268771, 272341, 275911, 305401, 363361, 365461, 388471, 498391, 516151, 556261, 561091, 585031, 752281, 776551, 783781, 812341, 832621, 911161, 942031, 950221, 1030021, 1108561

Offset: 1

Labos Elemer, Dec 19 2002

nonn

Equivalently, primes p such that p, p+6, p+10, p+12 and p+18 are consecutive primes.

			61 is in the sequence since 61, 67 = 61 + 6, 71 = 61 + 10, 73 = 61 + 12 and 79 = 61 + 18 are consecutive primes.

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

Subsequence of A078855. - R. J. Mathar, May 06 2017

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

Select[Partition[Prime[Range[50000]], 5, 1], Differences[#] == {6,4,2,6} &][[;;, 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 == 4 && p4 - p3 == 2 && p5 - p4 == 6, 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

A290635 Greatest of 4 consecutive primes with consecutive gaps 6, 4, 2.

Original entry on oeis.org

43, 73, 283, 619, 1303, 1669, 1789, 1873, 1999, 2143, 2383, 2689, 2803, 4519, 5419, 5443, 5653, 7879, 9013, 11833, 13693, 14563, 17389, 18133, 18313, 20359, 21493, 22159, 24109, 27283, 32719, 35533, 36793, 37573, 41233, 41959, 42409, 42463, 44269, 47149, 50593, 55219, 55819, 55933

Offset: 1

Muniru A Asiru, Aug 08 2017

nonn

All terms = {13, 19} mod 30.

			43 is a member of the sequence because 43 is the greatest of the 4 consecutive primes 31, 37, 41, 43 with consecutive gaps 6, 4, 2; that is, 37 - 31 = 6, 41 - 37 = 4, 43 - 41 = 2.

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

Subsequence of A006512 and A098413.

Cf. A022005, A078855.

K:=2*10^5+1;; # to get all terms <= K.
P:=Filtered([1,3..K],IsPrime);;  I:=Reversed([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]]);;
P3:=List(Positions(P2,I),i->P[i+Length(I)]);
# More efficient

Filtered(Set(Flat(List([13,19],j->List([1..2000],i->30*i+j)))),j->IsPrime(j) and IsPrime(j-12) and not IsPrime(j-10) and not IsPrime(j-8) and IsPrime(j-6) and not IsPrime(j-4) and IsPrime(j-2)); # Muniru A Asiru, Jul 03 2018

for i from 1 to 10^5 do if ithprime(i+1)=ithprime(i)+6 and ithprime(i+2)=ithprime(i)+4 and ithprime(i+3)=ithprime(i)+2  then print(ithprime(i+3)); fi; od; # Corrected by Robert Israel, Jun 28 2018
# More efficient:
primes:= select(isprime,[seq(seq(30*i+j,j=[13,19]),i=1..10^4)]):
select(t -> isprime(t-2) and isprime(t-6) and isprime(t-12) and not isprime(t-8), primes); # Robert Israel, Jun 28 2018

With[{s = Differences@ Prime@ Range[10^4]}, Prime[1 + SequencePosition[s, {6, 4, 2}][[All, -1]] ] ] (* Michael De Vlieger, Aug 16 2017 *)
Select[Partition[Prime[Range[6000]],4,1],Differences[#]=={6,4,2}&][[All,4]] (* Harvey P. Dale, Feb 13 2022 *)

is(n) = if(!ispseudoprime(n), return(0), my(v=[n-2, n-6, n-12]); if(v[1]==precprime(n-1) && v[2]==precprime(v[1]-1) && v[3]==precprime(v[2]-1), return(1))); 0 \\ Felix Fröhlich, Aug 10 2017

a(n) = A078855(n) + 12.

A385035 Primes p such that p + 8, p + 14, p + 18 and p + 20 are also primes.

Original entry on oeis.org

23, 53, 89, 263, 599, 1283, 1979, 3449, 5399, 5639, 11813, 14543, 41213, 42443, 44249, 47129, 55799, 57773, 65699, 74699, 75983, 79613, 84299, 87539, 88643, 88793, 88799, 113153, 115763, 126473, 143813, 148913, 150203, 160073, 163973, 167099, 176489, 178799, 178889, 209249

Offset: 1

Alexander Yutkin, Jun 15 2025

nonn

			p=23: 23+8=31, 23+14=37, 23+18=41, 23+20=43 —> prime quintuple: (23, 31, 37, 41, 43).

Cf. A000040.

Cf. A172454 [2, 4, 6], A078855 [6, 4, 2], A187057 [2, 4, 6, 8].

[p: p in PrimesUpTo(300000) | IsPrime(p+8) and IsPrime(p+14) and IsPrime(p+18) and IsPrime(p+20)]; // Vincenzo Librandi, Jul 04 2025

q:= p-> andmap(i-> isprime(p+i), [0, 8, 14, 18, 20]):
select(q, [5+6*i$i=0..35000])[];  # Alois P. Heinz, Jun 16 2025

Select[Prime[Range[20000]], AllTrue[#+{8, 14, 18,20}, PrimeQ]&] (* Stefano Spezia, Jun 18 2025 *)

cp's OEIS Frontend

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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A078961 Primes p such that the differences between the 5 consecutive primes starting with p are (6,4,2,4).

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A078962 Primes p such that the differences between the 5 consecutive primes starting with p are (6,4,2,6).

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A290635 Greatest of 4 consecutive primes with consecutive gaps 6, 4, 2.

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A385035 Primes p such that p + 8, p + 14, p + 18 and p + 20 are also primes.

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