A052188 -id:A052188 - cp's OEIS frontend

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

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].

A033447 Initial prime in set of 4 consecutive primes with common difference 12.

Original entry on oeis.org

111497, 258527, 286777, 318407, 332767, 341827, 358447, 439787, 473887, 480737, 495377, 634187, 647417, 658367, 663857, 703837, 732497, 816317, 819787, 827767, 843067, 862307, 937777, 970457, 970537, 1001267, 1012147, 1032727, 1052707, 1055827, 1104307, 1117877, 1164817, 1165837

Offset: 1

Jeff Burch

nonn

From Zak Seidov, Sep 30 2014: (Start)
All terms are == {7, 17} mod 30. There is no set of 5 consecutive primes in arithmetic progression with common difference 12 (because a(n)+48 is always divisible by 5).
Minimal first difference a(n+1)-a(n) = 40, and this occurs first at a(709) = 26930767, a(11357) = 655389367 and a(23339) = 1510368877; all a(n) are == 7 mod 30. (End)

Analogous sequences (start of CPAP-4 with common difference in square brackets): A033451 [6], this sequence [12], A033448 [18], A052242 [24], A052243 [30], A058252 [36], A058323 [42], A067388 [48], A259224 [54], A210683 [60].
Subsequence of A052188 and of A248085. - Zak Seidov, Jun 27 2015
Also subsequence of A054800: start of a CPAP-4, any common difference.

A033447 = Reap[For[p = 2, p < 1100000, p = NextPrime[p], p2 = NextPrime[p]; If[p2 - p == 12, p3 = NextPrime[p2]; If[p3 - p2 == 12, p4 = NextPrime[p3]; If[p4 - p3 == 12, Sow[p]]]]]][[2, 1]] (* Jean-François Alcover, Jun 28 2012 *)
Transpose[Select[Partition[Prime[Range[160000]],4,1],Union[ Differences[#]] =={12}&]][[1]] (* Harvey P. Dale, Jun 17 2014 *)

A033447(n, p=2, show_all=1, g=12,c,o)={forprime(q=p+1,, if(p+g!=p=q, next, q!=o+2*g, c=2, c++>3, show_all&& print1(o-g", "); n--||break); o=q-g); o-g} \\ Can be used as next(p)=A033447(1, p+1) to get the next term, e.g.:
p=0; A033447_vec=vector(30,i,p=A033447(1,p+1)) \\ M. F. Hasler, Oct 26 2018

More terms from Labos Elemer, Jan 31 2000

Definition clarified by Harvey P. Dale, Jun 17 2014

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

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

A052190 Primes p such that p, p+24, p+48 are consecutive primes.

Original entry on oeis.org

16763, 40039, 42509, 96353, 98573, 104183, 119243, 123863, 160093, 161783, 169259, 181789, 185243, 208529, 209719, 232753, 235699, 243343, 246049, 260339, 261799, 270073, 295363, 295703, 302459, 315199, 331399, 362003, 364079, 373669, 380729, 381793, 385943, 414809

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)=24."

			40039 is followed by 40063 and 40087, consecutive primes with equal distance of 24.

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

Subsequence of A098974.

Cf. A001223, A033451, A047948, A052188, A052189.

Select[Partition[Prime[Range[40000]],3,1],Differences[#]=={24,24}&][[All,1]] (* Harvey P. Dale, May 09 2019 *)

list(lim) = {my(p1 = 2, p2 = 3); forprime(p3 = 5, lim, if(p2 - p1 == 24 && p3 - p2 == 24, print1(p1, ", ")); p1 = p2; p2 = p3);} \\ Amiram Eldar, Feb 28 2025

Name changed by Jon E. Schoenfield, May 30 2018

A052187 a(n) is the smallest prime p such that p, p+d, and p+2d are consecutive primes where d = 2 for n = 1 and d = 6*(n-1) for n > 1.

Original entry on oeis.org

3, 47, 199, 20183, 16763, 69593, 255767, 247099, 3565931, 6314393, 4911251, 12012677, 23346737, 43607351, 34346203, 36598517, 51041957, 460475467, 652576321, 742585183, 530324329, 807620651, 2988119207, 12447231761, 383204539, 4470607951, 5007182707

Offset: 1

Labos Elemer, Jan 28 2000

nonn

The first term 3 is anomalous since for all others d is divisible by 6. These are minimal terms if in A047948 d=6 is replaced by possible differences: (2), 6, 12, 18, ..., 54, 60.

a(54) > 5*10^13, while a(55) = 46186474937633. - Giovanni Resta, Apr 08 2013

			a(2)=47 and it is the lower border of a dd pattern: 47[6 ]53[6 ]59. a(10)=6314393 and a(10)+54=6314447, a(10)+108=6314501 are consecutive primes and 6314393 is the smallest prime prior to a (54,54) difference pattern of A001223.

Jerry M Lagrou, Table of n, a(n) for n = 1..72 (terms 1..39 from Donovan Johnson, terms 40..53 from Giovanni Resta)

Cf. A001223, A047948, A052160, A054342.

Cf. A052188, A052189, A052195, A052196, A052197, A052198.

a = Table[0, {100}]; NextPrime[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; p = q = r = 0; Do[r = NextPrime[r]; If[r + p == 2q && r - q < 201 && a[[(r - q)/2]] == 0, a[[(r - q)/2]] = p]; p = q; q = r, {n, 1, 10^6}]; a (* Typos fixed by Zak Seidov, May 01 2020 *)

list(n)=ve=vector(n);ppp=2;pp=3;forprime(p=5,,d=p-pp;if(pp-ppp==d,i=d\6+1;if(i<=n&&ve[i]==0,ve[i]=ppp;print1(".");vecprod(ve)>0&&return(ve)));ppp=pp;pp=p) \\ Jeppe Stig Nielsen, Apr 17 2022

The least prime(k) such that prime(k+1) = (prime(k) + prime(k+2))/2 and prime(k+1) - prime(k) = d is either 2 or divisible by 6.

a(1) = A054342(1) - 2. For n>1, a(n) = A054342(n) - 6*(n-1). - Jeppe Stig Nielsen, Apr 16 2022

More terms from Labos Elemer, Jan 04 2002
More terms from Robert G. Wilson v, Jan 06 2002
Definition clarified by Harvey P. Dale, Aug 29 2012
a(23)-a(27) from Donovan Johnson, Aug 30 2012
Name edited by Jon E. Schoenfield, Nov 30 2023

A053072 Primes p such that p-12, p and p+12 are consecutive primes.

Original entry on oeis.org

211, 1511, 4409, 4691, 7841, 9871, 11299, 11411, 11731, 12841, 15161, 16619, 17431, 17851, 18341, 18731, 19739, 19949, 20161, 20521, 20731, 21661, 22051, 22259, 23801, 25621, 26041, 28069, 29599, 30059, 31051, 32479, 34171, 35129

Offset: 1

Harvey P. Dale, Feb 25 2000

In other words, balanced primes separated from the next lower and next higher prime neighbors by 12.

			1511 is separated from both the next lower prime and the next higher prime by 12.

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 211, p. 61, Ellipses, Paris, 2008.

Zak Seidov, Table of n, a(n) for n = 1..10000

Cf. A052188.

for i from 1 by 1 to 5000 do if ithprime(i+1) = ithprime(i) +12 and ithprime(i+2) = ithprime(i) + 24 then print(ithprime(i+1)); # Zerinvary Lajos, May 04 2007

lst={};Do[p=Prime[n];If[p-Prime[n-1]==Prime[n+1]-p==6*2,AppendTo[lst,p]],{n,2,2*7!}];lst (* Vladimir Joseph Stephan Orlovsky, May 20 2010 *)
Transpose[Select[Partition[Prime[Range[4000]],3,1],Differences[#] == {12,12}&]][[2]] (* Harvey P. Dale, Apr 07 2013 *)

a(n) = A052188(n) + 12. - Michel Marcus, Jan 09 2015

Corrected by Jud McCranie, Jan 04 2001

Edited by N. J. A. Sloane, Jul 03 2008 at the suggestion of R. J. Mathar

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.

A224325 First of three consecutive primes in arithmetic progression with gap of 6n, and such that a(n) > a(n-1).

Original entry on oeis.org

47, 199, 20183, 40039, 69593, 255767, 689467, 3565931, 6314393, 9113263, 12012677, 23346737, 43607351, 69266033, 75138781, 324237847, 460475467, 652576321, 742585183, 747570079, 807620651, 2988119207, 12447231761

Offset: 1

M. F. Hasler, Apr 03 2013

nonn

Without the condition on monotonicity, this would be essentially the same as A052187, but there 255767 is followed by 247099, while monotonicity here gives 689467. Similarly, following a(9) = A052187(10) = 6314393 we have a(10) = 9113263, while A052187(11) = 4911251. The next term which is not matching is a(14) = 69266033 vs A052187(15) = 34346203. One may notice that the two terms differ approximately by a factor of 2.

			a(1) = A047948(1) = 47 is the least prime p(k) such that p(k+1) - p(k) = p(k+2) - p(k+1) = 6.
a(2) = A052188(1) = 199 is the least prime p(k) > 47 such that p(k+1) - p(k) = p(k+2) - p(k+1) = 12.
a(3) = A052189(1) = 20183 is the least prime p(k) > 199 such that p(k+1) - p(k) = p(k+2) - p(k+1) = 18.
a(4) = A052190(1) = 40039 is the least prime p(k) > 20183 such that p(k+1) - p(k) = p(k+2) - p(k+1) = 24.
a(5) = A052195(1) = 69593 is the least prime p(k) > 40039 such that p(k+1) - p(k) = p(k+2) - p(k+1) = 30.

Pierre CAMI, Table of n, a(n) for n = 1..23

Cf. A224324 (gaps of 30n).

g=6;o=2;forprime(p=2,,o+g==(o=p)||next;nextprime(p+1)==p+g||next;print1(p-g",");g+=6)

cp's OEIS Frontend

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

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A033447 Initial prime in set of 4 consecutive primes with common difference 12.

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

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

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A052190 Primes p such that p, p+24, p+48 are consecutive primes.

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A052187 a(n) is the smallest prime p such that p, p+d, and p+2d are consecutive primes where d = 2 for n = 1 and d = 6*(n-1) for n > 1.

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