A033448 -id:A033448 - cp's OEIS frontend

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

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

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

A033450 List of sets of four consecutive primes in arithmetic progression with common difference 18.

Original entry on oeis.org

74453, 74471, 74489, 74507, 76543, 76561, 76579, 76597, 132893, 132911, 132929, 132947, 182243, 182261, 182279, 182297, 202823, 202841, 202859, 202877, 297403, 297421, 297439, 297457, 358793, 358811, 358829, 358847, 485923, 485941, 485959, 485977

Offset: 1

Jeff Burch

This is a 4-column table read by rows.

For the initial primes from each set of four, see A033448.

a:=[];
for n from 1 to 50000 do
  p1:=ithprime(n);
  p2:=nextprime(p1);
if (p2-p1)=18 then
  p3:=nextprime(p2);
     if p3-p2=18 then
        p4:=nextprime(p3);
         if p4-p3 = 18 then a:=[op(a),p1,p2,p3,p4]; fi;
     fi;
fi;
od:
a; # N. J. A. Sloane, Nov 23 2017

Select[Partition[Prime[Range[41000]],4,1],Union[Differences[#]]=={18}&]// Flatten (* Harvey P. Dale, Nov 24 2017 *)

Confirmed (and last set of four completed) by N. J. A. Sloane, Nov 23 2017

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

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

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

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

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

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A033450 List of sets of four consecutive primes in arithmetic progression with common difference 18.

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

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

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