A067388 -id:A067388 - cp's OEIS frontend

A054800 First term of balanced prime quartets: p(m+1)-p(m) = p(m+2)-p(m+1) = p(m+3)-p(m+2).

251, 1741, 3301, 5101, 5381, 6311, 6361, 12641, 13451, 14741, 15791, 15901, 17471, 18211, 19471, 23321, 26171, 30091, 30631, 53611, 56081, 62201, 63691, 71341, 74453, 75521, 76543, 77551, 78791, 80911, 82781, 83431, 84431, 89101, 89381

Offset: 1

Henry Bottomley, Apr 10 2000

nonn

This sequence is infinite if Dickson's conjecture holds. - Charles R Greathouse IV, Apr 23 2011

This is actually the complete list of primes starting a CPAP-4 (set of 4 consecutive primes in arithmetic progression). It equals A033451 for a(1..24), but it contains a(25) = 74453 which starts a CPAP-4 with common difference 18 (the first one with a difference > 6) and therefore is not in A033451. - M. F. Hasler, Oct 26 2018

			a(1) = 251 = prime(54) = A000040(54) and prime(55) - prime(54) = prime(56)-prime(55) = 6. - _Zak Seidov_, Apr 23 2011

Zak Seidov and Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 4000 terms from Seidov)

Cf. A006562, A054801 .. A054840.
Cf. A006560 (first prime to start a CPAP-n).
Start of CPAP-4 with given common difference (in square brackets): A033451 [6], A033447 [12], A033448 [18], A052242 [24], A052243 [30], A058252 [36], A058323 [42], A067388 [48], A259224 [54], A210683 [60].

Select[Partition[Prime[Range[9000]],4,1],Length[Union[Differences[#]]] == 1&][[All,1]] (* Harvey P. Dale, Aug 08 2017 *)

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

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

Original entry on oeis.org

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

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

A052243 Initial prime in set of (at least) 4 consecutive primes in arithmetic progression with difference 30.

Original entry on oeis.org

642427, 1058861, 3431903, 4176587, 4560121, 4721047, 5072269, 5145403, 5669099, 5893141, 6248969, 6285047, 6503179, 6682969, 8545357, 8776121, 8778739, 9490571, 9836227, 9843019, 9843049, 10023787, 11697979, 12057919, 12340313, 12687119, 12794641, 12845849

Offset: 1

Labos Elemer, Jan 31 2000

nonn

Primes p such that p, p+30, p+60, p+90 are consecutive primes.

The analogous sequence for a CPAP-5 (at least five consecutive primes in arithmetic progression) with gap 30 does not have its own entry in the OEIS, but for over 500 terms it is identical to A059044. The CPAP-6 analog is A058362. - M. F. Hasler, Jan 02 2020

			642427, 642457, 642487, 642517 are consecutive primes, so 642427 is in the sequence.

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (terms n = 1..200 from M. F. Hasler)
OEIS wiki, Consecutive primes in arithmetic progression: CPAP with given gap, updated Jan. 2020.
Index entries for sequences related to primes in arithmetic progressions

Analogous sequences (start of CPAP-4 with common difference in square brackets): A033451 [6], A033447 [12], A033448 [18], A052242 [24], A052243 [this: 30], A058252 [36], A058323 [42], A067388 [48], A259224 [54], A210683 [60].
Subsequence of A052195 and of A054800 (start of CPAP-4 with any common difference).
See also A059044 (start of CPAP-5), A058362 (CPAP-6).

f:=func; a:=[]; for p in PrimesInInterval(2,13000000) do if  (f(p)-p eq 30) and (f(f(p))-p eq 60) and (f(f(f(p)))-p eq 90) then Append(~a,p); end if; end for; a; // Marius A. Burtea, Jan 04 2020

p := 2 : q := 3 : r := 5 : s := 7 : for i from 1 do if q-p = 30 and r-q=30 and s-r=30 then printf("%d,\n",p) ; fi ; p := q ; q := r ; r := s ; s := nextprime(r) ; od: # R. J. Mathar, Apr 12 2008

p=2; q=3; r=5; s=7; A052243 = Reap[For[i=1, i<1000000, i++, If[ q-p == 30 && r-q == 30 && s-r == 30 , Print[p]; Sow[p]]; p=q; q=r; r=s; s=NextPrime[r]]][[2, 1]] (* Jean-François Alcover, Jun 28 2012, after R. J. Mathar *)
Transpose[Select[Partition[Prime[Range[1100000]],4,1],Union[ Differences[#]] =={30}&]][[1]] (* Harvey P. Dale, Jun 17 2014 *)

A052243(n,p=2,print_all=0,g=30,c,o)={forprime(q=p+1,,if(p+g!=p=q, next, q!=o+2*g, c=2, c++>3, print_all&& print1(o-g","); n--||break); o=q-g);o-g} \\ optional 2nd arg specifies starting point, allows to define:
next_A052243(p)=A052243(1,p+1) \\ replacing older code from 2008. - M. F. Hasler, Oct 26 2018

A052243 = { A052195(n) | A052195(n+1) = A052195(n) + 30 }. - M. F. Hasler, Jan 02 2020

More terms from Harvey P. Dale, Nov 19 2000

Edited by N. J. A. Sloane, Apr 28 2008, at the suggestion of R. J. Mathar

A033448 Initial prime in set of 4 consecutive primes in arithmetic progression with common difference 18.

Original entry on oeis.org

74453, 76543, 132893, 182243, 202823, 297403, 358793, 485923, 655453, 735883, 759113, 780613, 797833, 849143, 1260383, 1306033, 1442173, 1531093, 1534153, 1586953, 1691033, 1717063, 1877243, 1945763, 1973633, 2035513, 2067083, 2216803, 2266993, 2542513, 2556803, 2565203, 2805773

Offset: 1

Jeff Burch

nonn

Up to n = 10^4, the smallest difference a(n+1) - a(n) is 60 and occurs at n = 8571. - M. F. Hasler, Oct 26 2018

Each term is congruent to 3 mod 10 (as noted by Zak Seidov in the SeqFan email list). This means the three following consecutive primes are always congruent to 1, 9, and 7 mod 10, respectively (i.e., final digits for these primes are 3, 1, 9, 7, in that order). There cannot be a set of 5 such consecutive primes because a(n) + 4*18 == 5 (mod 10) so is a multiple of 5. - Rick L. Shepherd, Mar 27 2023

			{74453, 74471, 74489, 74507} is the first such set of 4 consecutive primes with common difference 18, so a(1) = 74453.

Analogous sequences (start of CPAP-4 with common difference in square brackets): A033451 [6], A033447 [12], A033448 [this: 18], A052242 [24], A052243 [30], A058252 [36], A058323 [42], A067388 [48], A259224 [54], A210683 [60].

Cf. A033450, A052189.

A033448 = Reap[For[p = 2, p < 2100000, p = NextPrime[p], p2 = NextPrime[p]; If[p2 - p == 18, p3 = NextPrime[p2]; If[p3 - p2 == 18, p4 = NextPrime[p3]; If[p4 - p3 == 18, Sow[p]]]]]][[2, 1]] (* Jean-François Alcover, Jun 28 2012 *)
Transpose[Select[Partition[Prime[Range[160000]],4,1],Union[ Differences[ #]] == {18}&]][[1]] (* Harvey P. Dale, Jun 17 2014 *)

A033448(n,show_all=1,g=18,p=2,o,c)={forprime(q=p+1,,if(p+g!=p=q,next, q!=o+2*g, c=3, c++>4, print1(o-g","); n--||break); o=q-g);o-g} \\ Can be used as nxt(p)=A033448(1,,,p+1), e.g.: {p=0;vector(20,i,p=nxt(p))} or {p=0;for(i=1,1e4,write("b.txt",i" "nxt(p)))}. - M. F. Hasler, Oct 26 2018

More terms from Labos Elemer, Jan 31 2000
Definition clarified by Harvey P. Dale, Jun 17 2014
Example reflecting final digits given by Rick L. Shepherd, Mar 27 2023

A059044 Initial primes of sets of 5 consecutive primes in arithmetic progression.

Original entry on oeis.org

9843019, 37772429, 53868649, 71427757, 78364549, 79080577, 98150021, 99591433, 104436889, 106457509, 111267419, 121174811, 121174841, 168236119, 199450099, 203908891, 207068803, 216618187, 230952859, 234058871, 235524781, 253412317, 263651161, 268843033, 294485363, 296239787

Offset: 1

Harvey Dubner (harvey(AT)dubner.com), Dec 18 2000

nonn

Each set has a constant difference of 30, for all of the terms listed so far.
It is conjectured that there exist arbitrarily long sequences of consecutive primes in arithmetic progression. As of December 2000, the record is 10 primes.
The first CPAP-5 with common difference 60 starts at 6182296037 ~ 6e9, cf. A210727. This sequence consists of first members of pairs of consecutive primes in A054800 (see also formula): a(1..6) = A054800({1555, 4555, 6123, 7695, 8306, 8371}). Conversely, pairs of consecutive primes in this sequence yield a term of A058362, i.e., they start a sequence of 6 consecutive primes in arithmetic progression (CPAP-6): e.g., the nearby values a(12) = 121174811, a(13) = 121174841 = a(12) + 30 indicate such a term, whence A006560(6) = A058362(1) = a(12). The first CPAP-6 with common difference 60 starts at 293826343073 ~ 3e11, cf. A210727. Longer CPAP's must have common difference >= 210. - M. F. Hasler, Oct 26 2018
About 500 initial terms of this sequence are the same as for the sequence "First of 5 consecutive primes separated by gaps of 30". The first 10^4 terms of A052243 give 281 terms of this sequence (up to ~ 3.34e9) with the same formula as the one using A054800, but as the above comment says, this will miss terms beyond twice that range. - M. F. Hasler, Jan 02 2020

David Wells, The Penguin Dictionary of Curious and Interesting Numbers (Rev. ed. 1997), p. 181.

Zak Seidov, Table of n, a(n) for n = 1..241 (all terms up to 3*10^9)
Jens K. Andersen, The Largest Known CPAP's, updated Sept. 2018
OEIS wiki, Consecutive primes in arithmetic progression, updated Jan. 2020
Index entries for sequences related to primes in arithmetic progressions

Cf. A054800: start of 4 consecutive primes in arithmetic progression (CPAP-4).
Cf. A033451, A033447, A033448, A052242, A052243, A058252, A058323, A067388: start of CPAP-4 with common difference 6, 12, 18, ..., 48.
Cf. A052239: start of first CPAP-4 with common difference 6n.
Cf. A058362: start of 6 consecutive primes in arithmetic progression.
Cf. A006560: first prime to start a CPAP-n.

Select[Partition[Prime[Range[14000000]],5,1],Length[Union[ Differences[ #]]]==1&] (* Harvey P. Dale, Jun 22 2013 *)

A059044(n,p=2,c,g,P)={forprime(q=p+1,, if(p+g!=p+=g=q-p, next, q!=P+2*g, c=3, c++>4, print1(P-2*g,",");n--||break);P=q-g);P-2*g} \\ This does not impose the gap to be 30, but it happens to be the case for the first values. - M. F. Hasler, Oct 26 2018

Found by exhaustive search for 5 primes in arithmetic progression with all other intermediate numbers being composite.

A059044 = { A054800(i) | A054800(i+1) - A151800(A054800(i)) } with the nextprime function A151800(prime(k)) = prime(k+1) = prime(k) + A001223(k). - M. F. Hasler, Oct 27 2018, edited Jan 02 2020.

a(16)-a(22) from Donovan Johnson, Sep 05 2008
Reference added by Harvey P. Dale, Jun 22 2013
Edited (definition clarified, cross-references corrected and extended) by M. F. Hasler, Oct 26 2018

A052239 Smallest prime p in set of 4 consecutive primes in arithmetic progression with common difference 6n.

Original entry on oeis.org

251, 111497, 74453, 1397609, 642427, 5321191, 23921257, 55410683, 400948369, 253444777, 1140813701, 491525857, 998051413, 2060959049, 4480114337, 55140921491, 38415872947, 315392068463, 15162919459, 60600021611, 278300877401, 477836574947, 1486135570643

Offset: 1

Labos Elemer, Jan 31 2000

See also the less restrictive A054701 where the gaps are multiples 6n. - M. F. Hasler, Nov 06 2018

			a(5) = 642427, 642457, 642487, 642517 are the smallest consecutive primes with 3 consecutive gaps of 30, cf. A052243.
From _M. F. Hasler_, Nov 06 2018: (Start)
Other terms are also initial terms of corresponding sequences:
a(1) = 251 = A033451(1) = A054800(1), start of first CPAP-4 with common gap of 6,
a(2) = 111497 = A033447(1), start of first CPAP-4 with common gap of 12,
a(3) = 74453 = A033448(1) = A054800(25), first CPAP-4 with common gap of 18,
a(4) = 1397609 = A052242(1), start of first CPAP-4 with common gap of 24,
a(5) = 642427 = A052243(1) = A052195(16), first CPAP-4 with common gap of 30,
a(6) = 5321191 = A058252(1) = A161534(26), first CPAP-4 with common gap 36 = 6^2,
a(7) = 23921257 = A058323(1), start of first CPAP-4 with common gap of 42,
a(8) = 55410683 = A067388(1), start of first CPAP-4 with common gap of 48,
a(9) = 400948369 = A259224(1), start of first CPAP-4 with common gap of 54,
a(10) = 253444777 = A210683(1) = A089234(417), CPAP-4 with common gap of 60,
a(11) = 1140813701 = A287547(1), start of first CPAP-4 with common gap of 66,
a(12) = 491525857 = A287550(1), start of first CPAP-4 with common gap of 72,
a(13) = 998051413 = A287171(1), start of first CPAP-4 with common gap of 78,
a(14) = 2060959049 = A287593(1), start of first CPAP-4 with common gap of 84,
a(15) = 4480114337 = A286817(1) = A204852(444), common distance 90. (End)

Jerry M Lagrou, Table of n, a(n) for n = 1..42
Index entries for sequences related to primes in arithmetic progressions

Initial terms of sequences A033451, A033447, A033448, A052242, A052243, A058252, A058323, A067388, A259224, A210683.
Range is a subset of A054800: start of 4 consecutive primes in arithmetic progression (CPAP-4).
Cf. A054701: gaps are possibly distinct multiples of 6n (not CPAP's).

Transpose[Flatten[Table[Select[Partition[Prime[Range[2000000]],4,1], Union[ Differences[ #]] =={6n}&,1],{n,7}],1]][[1]] (* Harvey P. Dale, Aug 12 2012 *)

a(n, p=[2, 0, 0], d=6*[n, n, n])={while(p+d!=p=[nextprime(p[1]+1), p[1], p[2]], ); p[3]-d[3]} \\ after M. F. Hasler in A052243; Graziano Aglietti (mg5055(AT)mclink.it), Aug 22 2010, Corrected by M. F. Hasler, Nov 06 2018

A052239(n, p=2, c, o)={n*=6; forprime(q=p+1, , if(p+n!=p=q, next, q!=o+2*n, c=2, c++>3, break); o=q-n); o-n} \\ M. F. Hasler, Nov 06 2018

More terms from Labos Elemer, Jan 04 2002
a(7) corrected and more terms added by Graziano Aglietti (mg5055(AT)mclink.it), Aug 22 2010
a(15)-a(20) from Donovan Johnson, Oct 05 2010
a(21)-a(23) from Donovan Johnson, May 23 2011

A210683 Primes p such that p, p+60, p+120, p+180 are consecutive primes.

Original entry on oeis.org

253444777, 271386581, 286000489, 415893013, 475992773, 523294549, 620164949, 794689481, 838188877, 840725323, 846389227, 884106599, 884951807, 908725507, 941796223, 952288331, 971614151, 1002290693, 1003166771, 1006976797, 1053792359, 1097338313, 1163141201

Offset: 1

Zak Seidov, May 09 2012

nonn

Subsequence of A089234 which itself is a subsequence of A126771:
a(1) = 253444777 = A089234(417) = A126771(81526),
a(36) = 1998782563 = A089234(5579) = A126771(788920).

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

Cf. A089234, A126771.
Analogous sequences (start of CPAP-4, with common difference in square brackets): A033451 [6], A033447 [12], A033448 [18], A052242 [24], A052243 [30], A058252 [36], A058323 [42], A067388 [48], A259224 [54].
Cf. A054800: union of all sequences of this type (start of CPAP-4).

A210683(n, p=2, v=1, g=60, c, o)={forprime(q=p+1, , if(p+g!=p=q, next, q!=o+2*g, c=2, c++>3, v&& print1(o-g", "); n--||break); o=q-g); o-g} \\ Can be used as next(p)=A210683(1, p) to get the next term, e.g.:
p=0; A210683_vec=vector(10,i,p=A210683(1,p)) \\ Will take a long time! - M. F. Hasler, Oct 26 2018

A259224 Initial primes in sets of 4 consecutive primes with common gap 54.

Original entry on oeis.org

400948369, 473838319, 583946599, 678953059, 816604199, 972598819, 1136526949, 1466715139, 1475790529, 1499794999, 1502149559, 1610895679, 1643313869, 1673057219, 1686181579, 1845792019, 1867046639, 1907478889, 1992202439, 2011077869, 2030490479, 2207714969

Offset: 1

Zak Seidov, Jun 21 2015

nonn

All terms are == {19,29} mod 30.

Lars Blomberg, Table of n, a(n) for n = 1..10000 (The first 102 terms from Zak Seidov)

Start of CPAP-4 with given common difference (in square brackets): A033451 [6], A033447 [12], A033448 [18], A052242 [24], A052243 [30], A058252 [36], A058323 [42], A067388 [48], A259224 [this: 54], A210683 [60].

Subsequence of A054800: start of a CPAP-4 with arbitrary common difference.

A259224(n, p=2, v=1, g=54, c, o)={forprime(q=p+1, , if(p+g!=p=q, next, q!=o+2*g, c=2, c++>3, v&& print1(o-g", "); n--||break); o=q-g); o-g} \\ Can be used as next(p)=A259224(1,p+1) to get the next term, e.g.:
p=0; A259224_vec=vector(10,i,p=A259224(1,p+1)) \\ Will be slow! - M. F. Hasler, Oct 26 2018

A058362 Initial primes of sets of 6 consecutive primes in arithmetic progression.

Original entry on oeis.org

121174811, 1128318991, 2201579179, 2715239543, 2840465567, 3510848161, 3688067693, 3893783651, 5089850089, 5825680093, 6649068043, 6778294049, 7064865859, 7912975891, 8099786711, 9010802341, 9327115723, 9491161423, 9544001791, 10101930253, 10523406343, 13193702321

Offset: 1

Harvey Dubner (harvey(AT)dubner.com), Dec 18 2000

nonn

For all the terms listed so far, the common difference is equal to 30. These are the smallest such sets.
It is conjectured that there exist arbitrarily long sequences of consecutive primes in arithmetic progression. As of December 2000 the record is 10 primes.
All terms are congruent to 9 (mod 14). - Zak Seidov, May 03 2017
The first CPAP-6 with common difference 60 starts at 293826343073 ~ 3*10^11, cf. A210727. [With a slope of a(n)/n ~ 5*10^8 this would correspond to n ~ 600.] This sequence consists of first members of pairs of consecutive primes in A059044. Conversely, a pair of consecutive primes in this sequence starts a CPAP-7. This must have a common difference >= 210. As of today, the smallest known CPAP-7 starts at 382003672700092872707633 ~ 3.8*10^23, cf. Andersen link. - M. F. Hasler, Oct 27 2018
The common difference of 60 first occurs at a larger-than-expected prime. The first CPAP-6 with common difference 90 starts at 8560443932347. The first CPAP-6 with common difference 120 starts at 1925601119017087. - Jerry M Lagrou, Jan 01 2024

Zak Seidov, Table of n, a(n) for n = 1..102
Jens K. Andersen, The Largest Known CPAP's, updated Sep 2018.
OEIS Wiki, Consecutive primes in arithmetic progression, updated Jan 2020.
Index entries for sequences related to primes in arithmetic progressions

Cf. A006560: first prime to start a CPAP-n.
Cf. A033451, A033447, A033448, A052242, A052243, A058252, A058323, A067388: start of CPAP-4 with common difference 6, 12, 18, ..., 48.
Cf. A054800: start of 4 consecutive primes in arithmetic progression (CPAP-4).
Cf. A052239: starting prime of first CPAP-4 with common difference 6n.
Cf. A059044: starting primes of CPAP-5.
Cf. A210727: starting primes of CPAP-5 with common difference 60.

p=c=g=P=0;forprime(q=1,, p+g==(p+=g=q-p)|| next; q==P+2*g&& c++|| c=3; c>5&& print1(P-3*g,","); P=q-g) \\ M. F. Hasler, Oct 26 2018

Equals { A059044(i) | A059044(i+1) = A151800(A059044(i)) }, A151800 = nextprime. - M. F. Hasler, Oct 30 2018

Corrected by Jud McCranie, Jan 04 2001
a(11)-a(18) from Donovan Johnson, Sep 05 2008
Comment split off from Name (to clarify definition) by M. F. Hasler, Oct 27 2018

cp's OEIS Frontend

A054800 First term of balanced prime quartets: p(m+1)-p(m) = p(m+2)-p(m+1) = p(m+3)-p(m+2).

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

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

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A052243 Initial prime in set of (at least) 4 consecutive primes in arithmetic progression with difference 30.

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A033448 Initial prime in set of 4 consecutive primes in arithmetic progression with common difference 18.

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A059044 Initial primes of sets of 5 consecutive primes in arithmetic progression.

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A052239 Smallest prime p in set of 4 consecutive primes in arithmetic progression with common difference 6n.

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