A210683 -id:A210683 - 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

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

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

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

A210727 Primes p such that p, p+60, p+120, p+180, p+240 are consecutive primes.

Original entry on oeis.org

6182296037, 6675135377, 6798668171, 10301484257, 12665852879, 14922537067, 26348961209, 27009595127, 30321479693, 35572512473, 36938181239, 37962662791, 45320751701, 45999570191, 50772316757, 52628649973, 55745449033, 56425976891, 57984707603, 60553081499

Offset: 1

Zak Seidov, May 10 2012

nonn

Subsequence of A210683: a(1) = 6182296037 = A210683(146), a(2) = 6675135377 = A210683(166), a(3) = 6798668171 = A210683(175).

The minimal possible value of the first differences of a set of six consecutive primes in arithmetic progression is 30 (see A058362 for examples). - Jon E. Schoenfield, Jan 04 2024

Zak Seidov, Table of n, a(n) for n = 1..300

Cf. A058362, A089234, A126771, A210683.

A287547 Initial prime in set of 4 consecutive primes in arithmetic progression with difference 66.

Original entry on oeis.org

1140813701, 1314331181, 1729804331, 2615969891, 2765625631, 3827771821, 4266876641, 4348917061, 4700742041, 4845745831, 4877408441, 5311420901, 5395463741, 5409482081, 5693097391, 5816498981, 5902417331, 6173160871, 6692523011, 6914652461, 6960900641

Offset: 1

Zak Seidov, May 26 2017

nonn

Lars Blomberg, Table of n, a(n) for n = 1..10000

Analogous sequences [with 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].

More terms from Lars Blomberg, May 30 2017

A287550 Initial prime in set of 4 consecutive primes in arithmetic progression with difference 72.

Original entry on oeis.org

491525857, 1470227987, 2834347387, 4314407477, 4766711387, 6401372837, 6871241197, 8971400797, 10168905497, 11776429517, 11871902557, 14538547967, 14925896087, 15218517367, 15646776877, 15875854927, 17310026197, 17942416307, 18347931587, 19241492057, 19379888947

Offset: 1

Zak Seidov, May 26 2017

nonn

a(1)=491525857=A052239(12).

Chai Wah Wu, Table of n, a(n) for n = 1..100

Analogous sequences [with 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]. Cf. A052239.

from gmpy2 import is_prime, next_prime
A287550_list, p = [], 2
q, r, s = p+72, p+144, p+216
while s <= 10**10:
    np = next_prime(p)
    if np == q and is_prime(r) and is_prime(s) and next_prime(q) == r and next_prime(r) == s:
        A287550_list.append(p)
    p, q, r, s = np, np+72, np+144, np+216 # Chai Wah Wu, Jun 03 2017

a(8)-a(21) from Chai Wah Wu, Jun 03 2017

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

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

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A210727 Primes p such that p, p+60, p+120, p+180, p+240 are consecutive primes.

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A287547 Initial prime in set of 4 consecutive primes in arithmetic progression with difference 66.