A033451 -id:A033451 - cp's OEIS frontend

A052198 Primes p such that p, p+42, p+84 are consecutive primes.

247099, 689467, 1008617, 1629767, 1658627, 2024647, 2750999, 2811719, 2880907, 2921777, 3264449, 3295027, 3311317, 3365449, 3555269, 3668419, 4059229, 4412099, 4440529, 4549309, 4619357, 4690219, 4802947, 4955179, 5115259

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)=42.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..2000 from Zak Seidov)

Cf. A001223, A033451, A047948, A052197.

Select[Partition[Prime[Range[400000]],3,1],Differences[#]=={42,42}&][[All,1]] (* Harvey P. Dale, May 28 2017 *)

is_A052198(n)=nextprime(n+1)==n+42 && nextprime(n+43)==n+84 && isprime(n) \\ Charles R Greathouse IV, Jan 07 2013, typo corrected by M. F. Hasler, Jan 13 2013

New name from Charles R Greathouse IV, Jan 07 2013

A090835 Numbers n such that if p=prime(n), then p, p+6, p+12, p+18 are consecutive primes with p=6*k+5 for some k, where prime(n) denotes n-th prime.

Original entry on oeis.org

54, 709, 821, 1510, 1594, 1726, 1842, 2009, 2600, 2876, 5689, 6252, 7438, 7728, 8090, 8229, 8654, 9105, 9184, 9997, 10934, 11343, 11390, 14193, 14866, 15000, 16320, 16748, 16950, 17246, 18466, 19164, 19802, 20152, 21508, 21692, 22048, 22270, 22997, 23242, 25435, 25466

Offset: 1

Pierre CAMI, Dec 09 2003

			prime(54) = 251: 251, 257, 263, 269 are consecutive primes.

Cf. A000040, A033451, A090832, A090833, A090834, A090836, A090837, A090838, A090839.

PrimePi/@Transpose[Select[Partition[Prime[Range[50000]],4,1],Differences[#]=={6, 6, 6}&&Mod[#[[1]],6]==5&]][[1]] (* Metin Sariyar, Sep 21 2019 *)

isok(n) = {my(p=prime(n), q, r, s); ((p % 6) == 5) && ((q=nextprime(p+1)) == p+6) && ((r=nextprime(q+1)) == p+12) && ((s=nextprime(r+1)) == p+18);} \\ Michel Marcus, Sep 20 2019

a(9) corrected and more terms from Michel Marcus, Sep 20 2019

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

A078969 Primes p such that the differences between the 5 consecutive primes starting with p are (6,6,6,4).

Original entry on oeis.org

3301, 15901, 18211, 30091, 53611, 71341, 77551, 80911, 89101, 120811, 252151, 285451, 292471, 294781, 344251, 601801, 616501, 744811, 792691, 809821, 908521, 912391, 1152631, 1154221, 1279801, 1376491, 1398031, 1455361, 1464271, 1500511, 1503031, 1555111, 1594261

Offset: 1

Labos Elemer, Dec 19 2002

nonn

Equivalently, primes p such that p, p+6, p+12, p+18 and p+22 are consecutive primes.

			30091 is in the sequence since 30091, 30097 = 30091 + 6, 30103 = 30091 + 12, 30109 = 30091 + 18 and 30113 = 30091 + 22 are consecutive primes.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Subsequence of A033451. - R. J. Mathar, May 06 2017

Cf. A001223, A078866, A078867, A078946-A078971, A022006, A022007.

Select[Partition[Prime[Range[150000]], 5, 1], Differences[#] == {6,6,6,4} &][[;;, 1]] (* Amiram Eldar, Feb 22 2025 *)

list(lim) = {my(p1 = 2, p2 = 3, p3 = 5, p4 = 7); forprime(p5 = 11, lim, if(p2 - p1 == 6 && p3 - p2 == 6 && p4 - p3 == 6 && p5 - p4 == 4, print1(p1, ", ")); p1 = p2; p2 = p3; p3 = p4; p4 = p5);} \\ Amiram Eldar, Feb 22 2025

a(n) == 1 (mod 30). - Amiram Eldar, Feb 22 2025

Edited by Dean Hickerson, Dec 20 2002

A382810 Primes p such that p + 6, p + 10 and p + 16 are also primes.

Original entry on oeis.org

7, 13, 31, 37, 73, 97, 157, 223, 373, 433, 1087, 1291, 1423, 1483, 1543, 1861, 1987, 2341, 2383, 2677, 2683, 3313, 3607, 4441, 4507, 4783, 4993, 5641, 5851, 6037, 6961, 7237, 7867, 8731, 9613, 9733, 10723, 13093, 13681, 14143, 14731, 16057, 16411, 16921, 17377

Offset: 1

Alexander Yutkin, Apr 05 2025

nonn

The four primes need not be consecutive; otherwise we have the sequence A078856.

			p=37: 37+6=43, 37+10=47, 37+16=53 -> prime quartet: (37, 43, 47, 53).

Robert Israel, Table of n, a(n) for n = 1..2500

Cf. A000040, A001223, A023200, A031924.

Cf. A078852 [4, 6, 6], A078856 [6, 4, 6], A078858 [6, 6, 4], A033451 [6, 6, 6].

q:= p-> andmap(i->isprime(p+i), [0, 6, 10, 16]):
select(q, [$2..20000])[];  # Alois P. Heinz, Apr 05 2025

Select[Prime[Range[2000]],AllTrue[#+{6,10,16},PrimeQ]&] (* James C. McMahon, Apr 13 2025 *)

A173892 Numbers k such that k and k+6 are both balanced primes.

Original entry on oeis.org

257, 1747, 3307, 5107, 5387, 6317, 6367, 12647, 13457, 14747, 15797, 15907, 17477, 18217, 19477, 23327, 26177, 30097, 30637, 53617, 56087, 62207, 63697, 71347, 75527, 77557, 78797, 80917, 82787, 83437, 84437, 89107, 89387, 91297, 94427, 95267

Offset: 1

Juri-Stepan Gerasimov, Mar 01 2010

nonn

k-6, k, k+6, and k+12 are consecutive primes.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A006562.

Cf. A054801. [From R. J. Mathar, Mar 29 2010]

a(n) = A033451(n) + 6.

Corrected and rewritten by Charles R Greathouse IV, Mar 19 2010

A078968 Primes p such that the differences between the 5 consecutive primes starting with p are (6,6,6,2).

Original entry on oeis.org

251, 17471, 56081, 75521, 94421, 115751, 121001, 154061, 163841, 179801, 185051, 250031, 344231, 351041, 380441, 417941, 517061, 683681, 703211, 713171, 783131, 849581, 916451, 983771, 1003091, 1025261, 1055591, 1070411, 1115561, 1129841, 1260881, 1517921, 1565171

Offset: 1

Labos Elemer, Dec 19 2002

nonn

Equivalently, primes p such that p, p+6, p+12, p+18 and p+20 are consecutive primes.

			251 is in the sequence since 251, 257 = 251 + 6, 263 = 251 + 12, 269 = 251 + 18 and 271 = 251 + 20 are consecutive primes.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Subsequence of A033451. - R. J. Mathar, May 06 2017

Cf. A001223, A078866, A078867, A078946-A078971, A022006, A022007.

Select[Partition[Prime[Range[150000]], 5, 1], Differences[#] == {6,6,6,2} &][[;;, 1]] (* Amiram Eldar, Feb 22 2025 *)

list(lim) = {my(p1 = 2, p2 = 3, p3 = 5, p4 = 7); forprime(p5 = 11, lim, if(p2 - p1 == 6 && p3 - p2 == 6 && p4 - p3 == 6 && p5 - p4 == 2, print1(p1, ", ")); p1 = p2; p2 = p3; p3 = p4; p4 = p5);} \\ Amiram Eldar, Feb 22 2025

a(n) == 11 (mod 30). - Amiram Eldar, Feb 22 2025

Edited by Dean Hickerson, Dec 20 2002

A196938 a(n) is the prime number that is the center element (3rd) of a 5-terms arithmetic progression prime chain.

Original entry on oeis.org

17, 29, 67, 71, 89, 97, 101, 131, 163, 167, 173, 191, 193, 197, 211, 233, 241, 257, 263, 269, 283, 307, 313, 317, 337, 347, 373, 419, 433, 443, 449, 457, 461, 463, 467, 479, 491, 503, 509, 521, 523, 547, 577, 599, 601, 607, 617, 619, 631, 641, 643, 677, 683

Offset: 1

Lei Zhou, Oct 07 2011

The Mathematica program gives the first 53 terms.

The Mathematica program is also good for finding sequences with any odd number of terms.

			{5,13,[17],23,29} is a 5-term arithmetic progression prime chain, so a(1)=17; (for all primes smaller than 17, no such chains exist)
{5,17,[29].41,53} is a 5-term arithmetic progression prime chain, so a(2)=29; (for all primes in between 17 and 29, no such chains exist)

Lei Zhou, Table of n, a(n) for n = 1..10000

Cf. A033451.

terms = 5; max=53; i = 1; step = (terms - 1)/2; Table[While[i++; p = Prime[i]; found = 0; j = 0; While[j++; df = 6*j; (p > (step*df)) && (found == 0), found = 1; Do[If[(! PrimeQ[p - k*df]) || (! PrimeQ[p + k*df]), found = 0], {k,1, step}]]; found == 0]; p, {ct, 1, max}]

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

A052198 Primes p such that p, p+42, p+84 are consecutive primes.

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A090835 Numbers n such that if p=prime(n), then p, p+6, p+12, p+18 are consecutive primes with p=6*k+5 for some k, where prime(n) denotes n-th prime.

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

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A078969 Primes p such that the differences between the 5 consecutive primes starting with p are (6,6,6,4).

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A382810 Primes p such that p + 6, p + 10 and p + 16 are also primes.

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Maple

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A173892 Numbers k such that k and k+6 are both balanced primes.

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A078968 Primes p such that the differences between the 5 consecutive primes starting with p are (6,6,6,2).

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A196938 a(n) is the prime number that is the center element (3rd) of a 5-terms arithmetic progression prime chain.