A033447 -id:A033447 - 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

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

A052242 Initial prime in set of 4 consecutive primes in arithmetic progression with common difference 24.

Original entry on oeis.org

1397609, 1436339, 2270459, 4181669, 4231919, 4425599, 4650029, 4967329, 7124099, 8254049, 8431369, 9000379, 9149639, 11343509, 11584009, 11734249, 12867319, 13723009, 14461229, 14590159, 15587659, 15776239, 15932899

Offset: 1

Labos Elemer, Jan 31 2000

nonn

Analogous sequences [with common difference in square brackets]: A033451 [6], A033447 [12], A033448 [18], A052242 [24], A052243 [30].

A052242 = Reap[For[p = 2, p < 16000000, p = NextPrime[p], p2 = NextPrime[p]; If[p2 - p == 24, p3 = NextPrime[p2]; If[p3 - p2 == 24, p4 = NextPrime[p3]; If[p4 - p3 == 24, Print[p]; Sow[p]]]]]][[2, 1]] (* Jean-François Alcover, Jun 28 2012 *)
Transpose[Select[Partition[Prime[Range[1100000]],4,1],Union[ Differences[#]] == {24}&]][[1]] (* Harvey P. Dale, Jun 17 2014 *)

More terms from Harvey P. Dale, Nov 25 2000

Definition clarified by Harvey P. Dale, Jun 17 2014

A058252 Initial prime in set of 4 consecutive primes with common difference 36.

Original entry on oeis.org

5321191, 8606621, 9148351, 41675791, 43251251, 49820291, 51825461, 57791281, 66637721, 73114441, 74055851, 82584841, 86801801, 87620011, 112161451, 123720361, 125810021, 126265751, 136413721, 140969291, 152777291, 153348161

Offset: 1

Harvey P. Dale, Dec 05 2000

nonn

Subsequence of A052197. - R. J. Mathar, Apr 12 2008
There are no 5 consecutive primes with common gap 36. - Zak Seidov, Jan 17 2013
If the primes are not required to be consecutive, the sequence starts 31, 241, 281, 311, 751, 911, 941, 1151, 1451, 2621, 4021, ... - Michael B. Porter, Jan 17 2013

Analogous sequences [with common difference in square brackets]: A033451 [6], A033447 [12], A033448 [18], A052242 [24], A052243 [30]

Transpose[Select[Partition[Prime[Range[8700000]],4,1], Union[ Differences[#]] =={36}&]][[1]]

a(16)-a(22) from Donovan Johnson, Sep 05 2008

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

A058323 Initial prime in set of 4 consecutive primes with common gap 42.

Original entry on oeis.org

23921257, 32611897, 33215597, 35650007, 44201617, 49945837, 51616717, 70350487, 70687937, 74816107, 78789707, 86066047, 99641917, 101568287, 129031187, 146922077, 149568217, 151779517, 153921017, 156793337, 162881627

Offset: 1

Harvey P. Dale, Dec 12 2000

nonn

All a(n) == 7 mod 10. - Robert Israel, May 13 2015

Analogous sequences (with differences in square brackets): A033451 [6], A033447 [12], A033448 [18], A052242 [24], A052243 [30], A058252 [36].

d[x_] := Prime[x+1]-Prime[x] {k1=42, k2=42, k3=42} k=0 Do[If[Equal[d[n], k1]&&Equal[d[n+1], k2]&& Equal[d[n+2], k3], k=k+1; Print[{k, n, Prime[n], Prime[n+1], Prime[n+2], Prime[n+3]}]], {n, 1, 10000000}]
Transpose[Select[Partition[Prime[Range[9000000]], 4, 1], Union[Differences[#]]=={42}&]][[1]] (* Vincenzo Librandi, Jun 21 2015 *)

More terms from Labos Elemer, Jan 11 2002
Definition corrected by Robert Israel, May 13 2015
Definition edited by Zak Seidov, Jun 21 2015
Offset changed 0 -> 1 by Zak Seidov, Jun 21 2015

A067388 Initial prime in set of 4 consecutive primes with common gap 48.

Original entry on oeis.org

55410683, 102291263, 141430363, 226383163, 280064453, 457433213, 531290533, 542418463, 555695713, 582949903, 629444003, 664652203, 665813153, 777809113, 802919653, 852404053, 887653633, 894328243, 898734673, 979048313, 993517643

Offset: 1

Labos Elemer, Jan 21 2002

nonn

Chai Wah Wu, Table of n, a(n) for n = 1..1000 (terms 1..102 from Zak Seidov)
Index entries for sequences related to primes in arithmetic progressions

Analogous sequences (with differences in square brackets): A033451[6], A033447[12], A033448[18], A052242[24], A052243[30], A058252[36], A058323[42], this sequence[48].

Transpose[Select[Partition[Prime[Range[10000000]], 4, 1], Union[Differences[#]]=={48}&]][[1]] (* Vincenzo Librandi, Jun 21 2015 *)

from sympy import isprime, nextprime
A067388_list, p = [], 2
q, r, s = p+48, p+96, p+144
while s <= 10**10:
    np = nextprime(p)
    if np == q and isprime(r) and isprime(s) and nextprime(q) == r and nextprime(r) == s:
        A067388_list.append(p)
    p, q, r, s = np, np+48, np+96, np+144 # Chai Wah Wu, Jun 01 2017

a(7)-a(21) from Donovan Johnson, Sep 05 2008

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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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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A052242 Initial prime in set of 4 consecutive primes in arithmetic progression with common difference 24.

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A058252 Initial prime in set of 4 consecutive primes with common difference 36.

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