A047948 -id:A047948 - cp's OEIS frontend

A033451 Initial prime in set of 4 consecutive primes with common difference 6.

251, 1741, 3301, 5101, 5381, 6311, 6361, 12641, 13451, 14741, 15791, 15901, 17471, 18211, 19471, 23321, 26171, 30091, 30631, 53611, 56081, 62201, 63691, 71341, 75521, 77551, 78791, 80911, 82781, 83431, 84431, 89101, 89381, 91291, 94421

Offset: 1

Jeff Burch

nonn

Primes p such that p, p+6, p+12, p+18 are consecutive primes.
It is conjectured that there exist arbitrarily long sequences of consecutive primes in arithmetic progression. As of March 2013 the record is 10 primes.
Note that the Green and Tao reference is about arithmetic progressions that are not necessarily consecutive. - Michael B. Porter, Mar 05 2013
Subsequence of A023271. - R. J. Mathar, Nov 04 2006
All terms p == 1 (mod 10) and hence p+24 are always divisible by 5. - Zak Seidov, Jun 20 2015
Subsequence of A054800, with which is coincides up to a(24), but a(25) = A054800(26). - M. F. Hasler, Oct 26 2018

			251, 257, 263, 269 are consecutive primes: 257 = 251 + 6, 263 = 251 + 12, 269 = 251 + 18.

T. D. Noe, Table of n, a(n) for n = 1..1000
Jens Kruse Andersen, The Largest Known CPAP's
Ben Green and Terence Tao, The primes contain arbitrarily long arithmetic progressions, arXiv:math/0404188 [math.NT], 2004-2007.
B. Green and T. Tao, The primes contain arbitrarily long arithmetic progressions, Annals of Math. 167(2008), 481-547.
OEIS wiki, Consecutive primes in arithmetic progression: CPAP with given gap, updated Jan. 2020
Index entries for sequences related to primes in arithmetic progressions

Intersection of A054800 and A023271.
Cf. A090832, A090833, A090834, A090835, A090836, A090837, A090838, A090839.
Analogous sequences [with common difference in square brackets]: A033447 [12], A033448 [18], A052242 [24], A052243 [30], A058252 [36], A058323 [42], A067388[48].
Cf. A058362, A059044.
Subsequence of A047948.

N:=10^5: # to get all terms <= N.
Primes:=select(isprime,[seq(i,i=3..N+18,2)]):
Primes[select(t->[Primes[t+1]-Primes[t], Primes[t+2]-Primes[t+1],
Primes[t+3]-Primes[t+2]]=[6,6,6], [$1..nops(Primes)-3])]; # Muniru A Asiru, Aug 04 2017

A033451 = Reap[ For[p = 2, p < 100000, p = NextPrime[p], p2 = NextPrime[p]; If[p2 - p == 6, p3 = NextPrime[p2]; If[p3 - p2 == 6, p4 = NextPrime[p3]; If[p4 - p3 == 6, Sow[p]]]]]][[2, 1]] (* Jean-François Alcover, Jun 28 2012 *)
Transpose[Select[Partition[Prime[Range[16000]],4,1],Union[ Differences[ #]] == {6}&]][[1]] (* Harvey P. Dale, Jun 17 2014 *)

p=2;q=3;r=5;forprime(s=7,1e4,if(s-p==18 && s-q==12 && s-r==6, print1(p", ")); p=q;q=r;r=s) \\ Charles R Greathouse IV, Feb 14 2013

a(n) = A000040(A090832(n)). - Zak Seidov, Jun 20 2015

A052378 Primes followed by a [4,2,4] prime difference pattern of A001223.

Original entry on oeis.org

7, 13, 37, 97, 103, 223, 307, 457, 853, 877, 1087, 1297, 1423, 1483, 1867, 1993, 2683, 3457, 4513, 4783, 5227, 5647, 6823, 7873, 8287, 10453, 13687, 13873, 15727, 16057, 16063, 16183, 17383, 19417, 19423, 20743, 21013, 21313, 22273, 23053, 23557

Offset: 1

Labos Elemer, Mar 22 2000

nonn

The sequence includes A052166, A052168, A022008 and also other primes like 13, 103, 16063 etc.
a(n) is the lesser term of a 4-twin (A023200) after which the next 4-twin comes in minimal distance [here it is 2; see A052380(4/2)].
Analogous prime sequences are A047948, A052376, A052377 and A052188-A052198 with various [d, A052380(d/2), d] difference patterns following a(n).
All terms == 1 (mod 6) - Zak Seidov, Aug 27 2012
Subsequence of A022005. - R. J. Mathar, May 06 2017

			103 initiates [103,107,109,113] prime quadruple followed by [4,2,4] difference pattern.

Zak Seidov, Table of n, a(n) for n = 1..2000

Cf. A023200, A053320, A022008, A052166, A052168, A001223, A052380.

a = {}; Do[If[Prime[x + 3] - Prime[x] == 10, AppendTo[a, Prime[x]]], {x, 1, 10000}]; a (* Zerinvary Lajos, Apr 03 2007 *)
Select[Partition[Prime[Range[3000]],4,1],Differences[#]=={4,2,4}&][[All,1]] (* Harvey P. Dale, Jun 16 2017 *)

is(n)=n%6==1 && isprime(n+4) && isprime(n+6) && isprime(n+10) && isprime(n) \\ Charles R Greathouse IV, Apr 29 2015

a(n) is the initial prime of a [p, p+4, p+6, p+6+4] prime-quadruple consisting of two 4-twins: [p, p+4] and [p+6, p+10].

A122535 Smallest prime of a triple of successive primes, where the middle one is the arithmetic mean of the other two.

Original entry on oeis.org

3, 47, 151, 167, 199, 251, 257, 367, 557, 587, 601, 647, 727, 941, 971, 1097, 1117, 1181, 1217, 1361, 1499, 1741, 1747, 1901, 2281, 2411, 2671, 2897, 2957, 3301, 3307, 3631, 3727, 4007, 4397, 4451, 4591, 4651, 4679, 4987, 5101, 5107, 5297, 5381, 5387

Offset: 1

Miklos Kristof, Sep 18 2006

nonn

Subsets are A047948, A052188, A052189, A052190, A052195, A052197, A052198, etc. - R. J. Mathar, Apr 11 2008
Could be generated by searching for cases A001223(i) = A001223(i+1), writing down A000040(i). - R. J. Mathar, Dec 20 2008
a(n) = A006562(n) - A117217(n). - Zak Seidov, Feb 12 2013
These are primes for which the subsequent prime gaps are equal, so (p(k+2)-p(k+1))/(p(k+1)-p(k)) = 1. It is conjectured that prime gaps ratios equal to one are less frequent than those equal to 1/2, 2, 3/2, 2/3, 1/3 and 3. - Andres Cicuttin, Nov 07 2016

			The prime 7 is not in the list, because in the triple (7, 11, 13) of successive primes, 11 is not equal (7 + 13)/2 = 10.
The second term, 47, is the first prime in the triple (47, 53, 59) of primes, where 53 is the mean of 47 and 59.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences related to primes in arithmetic progressions

Cf. A006562, A062839, A102552, A117217, A181424.

a122535 = a000040 . a064113  -- Reinhard Zumkeller, Jan 20 2012

Clear[d2, d1, k]; d2[n_] = Prime[n + 2] - 2*Prime[n + 1] + Prime[n]; d1[n_] = Prime[n + 1] - Prime[n]; k[n_] = -d2[n]/(1 + d1[n])^(3/2); Flatten[Table[If[k[n] == 0, Prime[n], {}], {n, 1, 1000}]] (* Roger L. Bagula, Nov 13 2008 *)
Transpose[Select[Partition[Prime[Range[750]], 3, 1], #[[2]] == (#[[1]] + #[[3]])/2 &]][[1]]  (* Harvey P. Dale, Jan 09 2011 *)

A122535()={n=3;ctr=0;while(ctr<50, avgg=( prime(n-2)+prime(n) )/2;
if( prime(n-1) ==avgg, ctr+=1;print( ctr,"  ",prime(n-2) )  );n+=1); } \\ Bill McEachen, Jan 19 2015

{A000040(i): A000040(i+1)= (A000040(i)+A000040(i+2))/2 }. - R. J. Mathar, Dec 20 2008

a(n) = A000040(A064113(n)). - Reinhard Zumkeller, Jan 20 2012

More terms from Roger L. Bagula, Nov 13 2008

Definition rephrased by R. J. Mathar, Dec 20 2008

A054643 Primes prime(n) such that prime(n) + prime(n+1) + prime(n+2) == 0 (mod 3).

Original entry on oeis.org

3, 47, 151, 167, 199, 251, 257, 367, 503, 523, 557, 587, 601, 647, 727, 941, 971, 991, 1063, 1097, 1117, 1181, 1217, 1231, 1361, 1453, 1493, 1499, 1531, 1741, 1747, 1753, 1759, 1889, 1901, 1907, 2063, 2161, 2281, 2393, 2399, 2411, 2441, 2671, 2897, 2957

Offset: 1

Labos Elemer, May 15 2000

nonn

The 2 differences of these 3 primes should be congruent of 6, except the first prime 3, for which 3 + 5 + 7 = 15 holds. Sequences like A047948, A052198 etc. are subsequences here.

			For prime(242) = 1531, the sum is 4623, the mean is 1541 and the successive differences are 6a=12 or 6b=6 resp.

Zak Seidov, Table of n, a(n) for n = 1..1000

Cf. A034961, A034707, A024675, A052288, A047948, A052198.
A122535 is a subsequence.
Cf. A075541 (for their indices).

Select[Partition[Prime@ Range@ 430, 3, 1], Divisible[Total@ #, 3] &][[All, 1]] (* Michael De Vlieger, Jun 29 2017 *)

A078857 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, 6,2]; short d-string notation of pattern = [662].

Original entry on oeis.org

47, 167, 257, 557, 587, 647, 1217, 2957, 4007, 6257, 6857, 7577, 10847, 11927, 14537, 16217, 17477, 19457, 24407, 25457, 26687, 26717, 29867, 41507, 41597, 48527, 51407, 54617, 56087, 60077, 61547, 68477, 75527, 82457, 84047, 94427, 101267

Offset: 1

Labos Elemer, Dec 11 2002

nonn

Subsequence of A047948. - R. J. Mathar, Feb 11 2013

			p=47,47+6=53,47+6+6=59,47+6+6+2=61 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].

Select[Partition[Prime[Range[10000]],4,1],Differences[#]=={6,6,2}&][[All,1]] (* Harvey P. Dale, Apr 29 2017 *)

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

Listed terms verified by Ray Chandler, Apr 20 2009

A078858 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, 6, 4]; short d-string notation of pattern = [664].

Original entry on oeis.org

151, 367, 601, 727, 2281, 2671, 3307, 4987, 5557, 10651, 12967, 13171, 15907, 18217, 18427, 20101, 20341, 24091, 27061, 28591, 30097, 30307, 31321, 32491, 35311, 37951, 41941, 42181, 42391, 45751, 52951, 53617, 55201, 56767, 59107, 65407

Offset: 1

Labos Elemer, Dec 11 2002

nonn

Subsequence of A047948. - R. J. Mathar, Feb 11 2013

			p=151, 151+6 = 157, 151+6+6 = 163, 151+6+6+4 = 167 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].

Transpose[Select[Partition[Prime[Range[6600]],4,1],Differences[#] == {6,6,4}&]][[1]] (* Harvey P. Dale, Nov 04 2011 *)

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

Listed terms verified by Ray Chandler, Apr 20 2009

A052188 Primes p such that p, p+12, p+24 are consecutive primes.

Original entry on oeis.org

199, 1499, 4397, 4679, 7829, 9859, 11287, 11399, 11719, 12829, 15149, 16607, 17419, 17839, 18329, 18719, 19727, 19937, 20149, 20509, 20719, 21649, 22039, 22247, 23789, 25609, 26029, 28057, 29587, 30047, 31039, 32467, 34159, 35117, 35839, 35899, 36217, 36809, 40099

Offset: 1

Labos Elemer, Jan 28 2000

nonn

Corresponds to two consecutive 12's in A001223. - - M. F. Hasler, Jan 02 2020

			a(1) = 199, followed by the consecutive primes 199 + 12 = 211, 199 + 12 + 12 = 223.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Zak Seidov)
OEIS wiki, Consecutive primes in arithmetic progression: CPAP with given gap, updated Jan. 2020.

Subsequence of A031930.
Generalization of A047948 and A033451 if 6 replaced by 12.
Cf. A001223, A033447.

[p:p in PrimesUpTo(36000)| NextPrime(p)-p eq 12 and  NextPrime(p+12)-p eq 24]; // Marius A. Burtea, Jan 03 2020

Transpose[Select[Partition[Prime[Range[3800]],3,1], Union[Differences[#]] =={12}&]][[1]]  (* Harvey P. Dale, Apr 26 2011 *)

lista(nn) = {forprime(p=1, nn, q = nextprime(p+1); r = nextprime(q+1); if ((r-q==12) && (q-p==12), print1(p, ", ")););} \\ Michel Marcus, Jun 27 2015

Name changed by Jon E. Schoenfield, May 30 2018

A052381 The smallest initial prime of 2 non-overlapping d-twin primes if the distance between pairs (D) is minimal (see A052380).

Original entry on oeis.org

3, 7, 47, 389, 409, 199, 24749, 3373, 20183, 46703, 19867, 16763, 142811, 14563, 69593, 763271, 276637, 255767, 363989, 383179, 247099, 2130809, 15370423, 3565931, 458069, 9401647, 6314393, 20823437, 9182389, 4911251, 15442121

Offset: 1

Labos Elemer, Mar 13 2000

nonn

A prime quadruple (triple), {[p,p+d],[p+D,p+D+d]} is called a "non-overlapping" (disjoint or touching) pair of twins if D = distance >= d = difference "inside" twin.

			If n=23, d=46, min{D}=48 then the first suitable quadruple of primes is [15370423, 15370469, 15370471, 15370517] with difference pattern [46, 2, 46]; if n=3, d=6, min{D}=6 then the first such triple is [47, 53, 53, 59] = [47, 53, 59] with difference pattern [6, 6].

The first 10 terms here appear as initial terms in A052350-A052359.

Smallest p so that [p, p+2n], [p+min{D}, p+2n+min{D}] is a quadruple (or triple if d=min{D}) of consecutive primes.

Corrected by Jud McCranie, Jan 04 2001

a(11) corrected by Sean A. Irvine, Nov 07 2021

A052195 Primes p such that p, p+30, p+60 are consecutive primes.

Original entry on oeis.org

69593, 110651, 134609, 228647, 237791, 250889, 303157, 318919, 396449, 421913, 498271, 507431, 535243, 554317, 629623, 642427, 642457, 668243, 692161, 716003, 729791, 780523, 782581, 790897, 801217, 825131, 829289, 847393, 892291, 902873, 940097, 942449, 963913, 995243, 1027067

Offset: 1

Labos Elemer, Jan 28 2000

nonn

			69593, 69623, 69653 are consecutive primes with equal distance d = 30.
110651, 110681 and 110711 are consecutive primes with equal distance d = 30.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Zak Seidov)
OEIS wiki, Consecutive primes in arithmetic progression: CPAP with given gap, updated Jan. 2020.

Subsequence of A124596 (primes followed by gap 30).
Cf. A047948 (analog for gap 6), A052188 (gap 12), A052189 (gap 18), A052190 (gap 24), A053075 (a(n) + 30).
Cf. A001223 (gaps), A052243 (quadruplets with gap 30), A033451 (quadruplets with gap 6).

Select[Partition[Prime[Range[80000]],3,1],Differences[#]=={30,30}&][[All,1]] (* Harvey P. Dale, May 03 2018 *)

vecextract(A124596, select(t->t==30, A124596[^1]-A124596[^-1],1)) \\ Terms of A124596 with indices of first differences of 30. Gives a(1..230) from A124596(1..10^4). - M. F. Hasler, Jan 02 2020

{ A124596(n) | A124596(n+1) = A124596(n) + 30 }. - M. F. Hasler, Jan 02 2020

A052189 Primes p such that p, p+18, p+36 are consecutive primes.

Original entry on oeis.org

20183, 21893, 25373, 29251, 30431, 34613, 50423, 54833, 56131, 58111, 63541, 66413, 74453, 74471, 76543, 76561, 77933, 78241, 81421, 107563, 108421, 110441, 112163, 121403, 122081, 122561, 131023, 132893, 132911, 135283, 137303, 137831, 143141, 144593, 145643

Offset: 1

Labos Elemer, Jan 28 2000

nonn

Old name was "Primes p(k) such that p(k+2)-p(k+1)=p(k+1)-p(k)=18."

			20183 is a term since , 20183, 20201, and 20219 are consecutive primes with difference of 18.

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

Subsequence of A031936
A033448 is a subsequence.
Cf. A001223, A033451, A047948, A052188.

Select[Partition[Prime[Range[15000]], 3, 1], Differences[#] == {18, 18} &][[;; , 1]] (* Amiram Eldar, Feb 28 2025 *)

list(lim) = {my(p1 = 2, p2 = 3); forprime(p3 = 5, lim, if(p2 - p1 == 18 && p3 - p2 == 18, print1(p1, ", ")); p1 = p2; p2 = p3);} \\ Amiram Eldar, Feb 28 2025

Name changed by Jon E. Schoenfield, May 30 2018

cp's OEIS Frontend

A033451 Initial prime in set of 4 consecutive primes with common difference 6.

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A052378 Primes followed by a [4,2,4] prime difference pattern of A001223.

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A122535 Smallest prime of a triple of successive primes, where the middle one is the arithmetic mean of the other two.

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A054643 Primes prime(n) such that prime(n) + prime(n+1) + prime(n+2) == 0 (mod 3).

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A078857 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, 6,2]; short d-string notation of pattern = [662].

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A078858 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, 6, 4]; short d-string notation of pattern = [664].

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A052188 Primes p such that p, p+12, p+24 are consecutive primes.

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