A078848 -id:A078848 - 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 *)

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

A233540 Primes p such that p+2, p+8, and p+12 are all prime.

Original entry on oeis.org

5, 11, 29, 59, 71, 101, 269, 431, 1289, 1481, 2129, 2339, 2381, 2789, 4721, 5519, 5639, 5849, 6569, 6959, 8999, 10091, 13679, 14549, 16061, 16649, 16691, 18119, 19379, 19421, 19751, 21011, 21491, 22271, 25931, 27689, 27791, 28619, 31181, 32369, 32561, 32831

Offset: 1

K. D. Bajpai, Dec 12 2013

nonn

The primes produced (p, p+2, p+8, p+12) are not always consecutive primes.

			29 is in the sequence because 29, 29 + 2 = 31, 29 + 8 = 37, and 29 + 12 = 41 are all prime.

Michael B. Porter, Table of n, a(n) for n = 1..2300

Cf. A007530 (prime quadruples).

Cf. A078848 (same prime differences, but with consecutive primes).

KD := proc() local a,b,c,p; p:=ithprime(n);a:=p+2;b:=p+8;c:=p+12;if isprime(a)and isprime(b) and isprime(c) then RETURN (p); fi; end: seq(KD(), n=1..10000);
# K. D. Bajpai, Dec 27 2013

Select[Prime[Range[4000]],AllTrue[#+{2,8,12},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Mar 04 2016 *)

is_a233540(p) = isprime(p) && isprime(p+2) && isprime(p+8) && isprime(p+12) \\ Michael B. Porter, Dec 27 2013

A046141 INTERSECT A046134. - R. J. Mathar, Aug 20 2019

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

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

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A233540 Primes p such that p+2, p+8, and p+12 are all prime.

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Maple

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