cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-8 of 8 results.

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

Views

Author

Jeff Burch

Keywords

Comments

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

Examples

			251, 257, 263, 269 are consecutive primes: 257 = 251 + 6, 263 = 251 + 12, 269 = 251 + 18.

Links

Crossrefs

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.

Programs

  • Maple
    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
  • Mathematica
    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 *)
  • PARI
    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

Formula

a(n) = A000040(A090832(n)). - Zak Seidov, Jun 20 2015

A090832 Numbers k such that p(k), p(k)+6, p(k)+12, p(k)+18 are consecutive primes, where p(k) denotes k-th prime.

Original entry on oeis.org

54, 271, 464, 682, 709, 821, 829, 1510, 1594, 1726, 1842, 1853, 2009, 2086, 2209, 2600, 2876, 3253, 3303, 5463, 5689, 6252, 6386, 7064, 7438, 7620, 7728, 7918, 8090, 8145, 8229, 8631, 8654, 8828, 9105, 9184, 9243, 9997, 10052, 10074, 10329, 10934, 11257, 11343
Offset: 1

Views

Author

Pierre CAMI, Dec 09 2003

Keywords

Examples

			p(271)=1741: 1741,1747,1753,1759 are consecutive primes,1747=1741+6,1753=1741+12,1759=1741+18

Links

Crossrefs

Cf. A033451, A090833, A090834, A090835, A090836, A090837, A090838, A090839.

Programs

  • Mathematica
    p[n_]:=Prime[n];Select[Range[15000],p[ #+1]-p[ # ]==p[ #+2]-p[ #+1]==p[ #+3]-p[ #+2]==6&] (* Zak Seidov, Mar 05 2006 *)
PrimePi[#[[1]]]&/@Select[Partition[Prime[Range[11000]],4,1],Differences[#]=={6,6,6}&] (* Harvey P. Dale, Oct 28 2023 *)

Extensions

Corrected and extended by Zak Seidov, Mar 05 2006

A090839 Numbers k such that 6*k+1, 6*k+7, 6*k+13, 6*k+19 are consecutive primes.

Original entry on oeis.org

290, 550, 850, 1060, 2650, 3035, 3245, 5015, 5105, 8935, 10615, 11890, 12925, 13485, 13905, 14850, 15215, 15985, 17560, 17600, 18105, 19925, 20135, 21780, 23510, 24040, 25490, 28830, 31145, 34365, 36355, 38140, 38370, 42025, 43845, 46820, 47575, 48745, 49130, 50495, 53350
Offset: 1

Views

Author

Pierre CAMI, Dec 09 2003

Keywords

Comments

All terms are == 0 (mod 5). - Robert G. Wilson v, Dec 12 2017

Examples

			6*290 + 1 = 1741, 6*290 + 7 = 1747, 6*290 + 13 = 1753, 6*290 + 19 = 1759 and 1741, 1747, 1753, 1759 are consecutive primes, so 290 is a term.

Links

Crossrefs

Cf. A033451, A090832, A090833, A090834, A090835, A090836, A090837, A090838.

Programs

  • Mathematica
    Block[{nn = 50500, s}, s = Select[Prime@ Range@ PrimePi[6 (nn + 3) - 1], Divisible[(# + 1), 6] &]; Select[Range@ nn, And[AllTrue[#, PrimeQ], Count[s, q_ /; First[#] < q < Last@ #] == 0] &@ Map[6 # + 1 &, # + Range[0, 3]] &]] (* Michael De Vlieger, Dec 06 2017 *)
fQ[n_] := Block[{p = {6n +1, 6n +7, 6n +13, 6n +19}}, Union@ PrimeQ@ p == {True} && NextPrime[6n +1, 3] == 6n +19]; Select[5 Range@ 10100, fQ] (* Robert G. Wilson v, Dec 12 2017 *)
Select[(#-1)/6&/@Select[Partition[Prime[Range[30000]],4,1],Differences[#]=={6,6,6}&][[;;,1]],IntegerQ] (* Harvey P. Dale, Apr 05 2025 *)
  • PARI
    isok(n) = my(p,q,r); isprime(p=6*n+1) && ((q=6*n+7) == nextprime(p+1)) && ((r=6*n+13) == nextprime(q+1)) && (6*n+19 == nextprime(r+1)); \\ Michel Marcus, Sep 20 2019

Extensions

Missing term 5105 and more terms from Michel Marcus, Sep 20 2019

A090836 Numbers n such that 6*n+5, 6*n+11, 6*n+17, 6*n+23 are consecutive primes.

Original entry on oeis.org

41, 896, 1051, 2106, 2241, 2456, 2631, 2911, 3886, 4361, 9346, 10366, 12586, 13131, 13796, 14071, 14896, 15736, 15876, 17451, 19291, 20091, 20166
Offset: 1

Views

Author

Pierre CAMI, Dec 09 2003

Keywords

Examples

			6*41+5=251, 6*41+11=257, 6*41+17=263, 6*41+23=269; 251,257,263,269 are consecutive primes.

Links

Crossrefs

Cf. A033451, A090832, A090833, A090834, A090835, A090837, A090838, A090839.

Programs

  • Mathematica
    Select[(#-5)/6&/@Transpose[Select[Partition[Prime[Range[11500]],4,1], Union[Differences[#]]=={6}&]][[1]],IntegerQ] (* Harvey P. Dale, Nov 18 2013 *)

A090833 Numbers n such that 6n+5, 6n+11, 6n+17, 6n+23 are consecutive primes or 6n+1, 6n+7, 6n+13, 6n+19 are consecutive primes.

Original entry on oeis.org

41, 290, 550, 850, 896, 1051, 1060, 2106, 2241, 2456, 2631, 2650, 2911, 3035, 3245, 3886, 4361, 5015, 5105, 8935, 9346, 10366, 10615, 11890, 12586, 12925, 13131, 13485, 13796, 13905, 14071, 14850, 14896, 15215, 15736, 15876, 15985, 17451, 17560
Offset: 1

Views

Author

Pierre CAMI, Dec 09 2003

Keywords

Links

Crossrefs

Cf. A033451, A090832, A090834, A090835, A090836, A090837, A090838, A090839.

Programs

  • Mathematica
    Select[Range[20000],(PrimeQ[6#+5]&&Differences[NestList[ NextPrime[ #]&, 6 #+5,3]] =={6,6,6})||(PrimeQ[6#+1]&&Differences[NestList[NextPrime[#]&,6 #+1,3]]=={6,6,6})&] (* Harvey P. Dale, Sep 23 2016 *)
  • PARI
    p=2;q=3;r=5;forprime(s=7,1e5,if(s-p==18&&s-q==12&&s-r==6,print1(p\6", "));p=q;q=r;r=s) \\ Charles R Greathouse IV, Dec 27 2011

Extensions

a(19) from Charles R Greathouse IV, Dec 27 2011

A090837 Primes p such that p, p+6, p+12, p+18 are consecutive primes and p = 6*k+1 for some k.

Original entry on oeis.org

1741, 3301, 5101, 6361, 15901, 18211, 19471, 30091, 30631, 53611, 63691, 71341, 77551, 80911, 83431, 89101, 91291, 95911, 105361, 105601, 108631, 119551, 120811, 130681, 141061, 144241, 152941, 172981, 186871, 206191, 218131, 228841, 230221, 252151, 263071, 280921, 285451
Offset: 1

Views

Author

Pierre CAMI, Dec 09 2003

Keywords

Examples

			1741, 1747, 1753, 1759 are consecutive primes, 1747 = 1741 + 6, 1753 = 1741 + 12, 1759 = 1741 + 18 and 1741 = 6 * 290 + 1.

Links

Crossrefs

Cf. A033451, A090832, A090833, A090834, A090835, A090836, A090838, A090839.

Programs

  • Maple
    filter:= p -> isprime(p) and nextprime(p) = p+6 and nextprime(p+6)=p+12 and nextprime(p+12)=p+18:
select(filter, [seq(i,i=1..10^6,6)]); # Robert Israel, Nov 11 2020
  • PARI
    isok(p) = my(q,r,s); isprime(p) && ((p % 6) == 1) && ((q=nextprime(p+1)) == p+6) && ((r=nextprime(q+1)) == p+12) && ((s=nextprime(r+1)) == p+18); \\ Michel Marcus, Sep 20 2019

Extensions

More terms from Michel Marcus, Sep 20 2019

A090838 Numbers n such that p(n),p(n)+6,p(n)+12,p(n)+18 are consecutive primes and p(n)=6*k+1 for some k, where p(n) denotes n-th prime.

Original entry on oeis.org

271, 464, 682, 829, 1853, 2086, 2209, 3253, 3303, 5463, 6386, 7064, 7620, 7918, 8145, 8631, 8828, 9243, 10052, 10074, 10329, 11257, 11368, 12223, 13100, 13359, 14105, 15751, 16909, 18481, 19455, 20332, 20456, 22213, 23071, 24510, 24874, 25420, 25595, 26233
Offset: 1

Views

Author

Pierre CAMI, Dec 09 2003

Keywords

Examples

			p(271)=1741: 1741,1747,1753,1759 are consecutive primes,1747=1741+6,1753=1741+12,1759=1741+18 and 1741=6*290+1

Links

Crossrefs

Cf. A033451, A090832, A090833, A090834, A090835, A090836, A090837, A090839.

Programs

  • Mathematica
    PrimePi/@Transpose[Select[Partition[Prime[Range[50000]],4,1], Differences[ #] == {6,6,6}&&Mod[#[[1]],6]==1&]][[1]] (* Harvey P. Dale, Nov 04 2015 *)

Extensions

More terms from Harvey P. Dale, Nov 04 2015

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

Views

Author

Pierre CAMI, Dec 09 2003

Keywords

Examples

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

Crossrefs

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

Programs

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

Extensions

a(9) corrected and more terms from Michel Marcus, Sep 20 2019
Showing 1-8 of 8 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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