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 1-10 of 22 results. Next

A046121 Duplicate of A023271.

Original entry on oeis.org

5, 11, 41, 61, 251, 601, 641, 1091, 1481, 1601, 1741, 1861, 2371, 2671, 3301, 3911
Offset: 1

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Author

Keywords

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

A227284 First primes of arithmetic progressions of 9 primes each with the common difference 210.

Original entry on oeis.org

199, 409, 3499, 10859, 564973, 1288607, 1302281, 2358841, 3600521, 4047803, 17160749, 20751193, 23241473, 44687567, 50655739, 53235151, 87662609, 100174043, 103468003, 110094161, 180885839, 187874017, 192205147, 221712811, 243051733, 243051943, 304570103
Offset: 1

Views

Author

Sameen Ahmed Khan, Jul 05 2013

Keywords

Comments

The minimal possible difference in an AP-k is conjectured to be k# for all k > 7.
When a(n+1) = a(n) + 210, as for n = 1, 25, ..., then a(n) is in A094220: start of AP of 10 primes with common distance 210. - M. F. Hasler, Jan 02 2020

Examples

			p = 409 then the AP-9 is {409, 619, 829, 1039, 1249, 1459, 1669, 1879, 2089} with the difference 9# = 2*3*5*7 = 210.

Links

Crossrefs

Cf. A001359, A023241, A023271, A094220, A156204, A227281, A227282, A227283, A227285, A227286.

Programs

  • Mathematica
    Clear[p]; d = 210; ap9p = {}; Do[If[PrimeQ[{p, p + d, p + 2*d, p + 3*d, p + 4*d, p + 5*d, p + 6*d, p + 7*d, p + 8*d}] == {True, True, True, True, True, True, True, True, True}, AppendTo[ap9p, p]], {p, 3, 10^9, 2}]; ap9p
  • PARI
    v=[1..8]*210; forprime(p=1,,for(i=1,#v,isprime(p+v[i])||next(2));print1(p",")) \\ M. F. Hasler, Jan 02 2020

A227281 First primes of arithmetic progressions of 5 primes each with the common difference 30.

Original entry on oeis.org

7, 11, 37, 107, 137, 151, 277, 359, 389, 401, 541, 557, 571, 877, 1033, 1493, 1663, 2221, 2251, 2879, 3271, 6269, 6673, 6703, 7457, 7487, 9431, 10103, 10133, 10567, 11981, 12457, 12973, 14723, 17047, 19387, 24061, 25643, 25673, 26861, 26891, 27337, 27367
Offset: 1

Views

Author

Sameen Ahmed Khan, Jul 05 2013

Keywords

Comments

The minimal possible difference in an AP-k is conjectured to be k# for all k > 7.
For k = 5, we have d = 3# = 6 and there is ONLY one AP-5 with this difference: {5, 11, 17, 23, 29}.

Examples

			p = 11 then {11, 11 + 1*30, 11 + 2*30, 11 + 3*30, 11 + 4*30} = {11, 41, 71, 101, 131}, which is 5 primes in arithmetic progression with the difference 5# = 30.

Links

Crossrefs

Cf. A001359, A023241, A023271, A094220, A156204, A227282, A227283, A227284, A227285, A227286.

Programs

  • Mathematica
    Clear[p]; d = 30; ap5p = {}; Do[If[PrimeQ[{p, p + d, p + 2*d, p + 3*d, p + 4*d}] == {True, True, True, True, True}, AppendTo[ap5p, p]], {p, 3, 25000, 2}]; ap5p

A046122 Second member of a sexy prime quadruple: value of p+6 such that p, p+6, p+12 and p+18 are all prime.

Original entry on oeis.org

11, 17, 47, 67, 257, 607, 647, 1097, 1487, 1607, 1747, 1867, 2377, 2677, 3307, 3917, 4007, 5107, 5387, 5437, 5647, 6317, 6367, 9467, 11827, 12107, 12647, 13457, 14627, 14747, 15797, 15907, 17477, 18217, 19477, 20347, 21487, 23327, 24097
Offset: 1

Views

Author

Eric W. Weisstein

Keywords

Links

  • Amiram Eldar, Table of n, a(n) for n = 1..10000
  • Eric Weisstein's World of Mathematics, Sexy Primes. [The definition in this webpage is unsatisfactory, because it defines a "sexy prime" as a pair of primes.- _N. J. A. Sloane_, Mar 07 2021].

Crossrefs

Cf. A023201, A023271, A046117, A046123, A046124.

Programs

  • Mathematica
    lst={};Do[p=Prime[n];If[PrimeQ[p+6]&&PrimeQ[p+12]&&PrimeQ[p+18], AppendTo[lst, p+6]], {n, 8!}];lst (* Vladimir Joseph Stephan Orlovsky, Aug 29 2008 *)

Formula

a(n) = 6 + A023271(n) = A046123(n) - 6. - R. J. Mathar, Jun 28 2012

A046123 Third member of a sexy prime quadruple: value of p+12 such that p, p+6, p+12 and p+18 are all prime.

Original entry on oeis.org

17, 23, 53, 73, 263, 613, 653, 1103, 1493, 1613, 1753, 1873, 2383, 2683, 3313, 3923, 4013, 5113, 5393, 5443, 5653, 6323, 6373, 9473, 11833, 12113, 12653, 13463, 14633, 14753, 15803, 15913, 17483, 18223, 19483, 20353, 21493, 23333, 24103
Offset: 1

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Author

Eric W. Weisstein

Keywords

Comments

Is 17 the only term that is not equal to 3 mod 10? It is the only such term up to the one millionth prime. - Harvey P. Dale, Jan 25 2023

Links

  • Amiram Eldar, Table of n, a(n) for n = 1..10000
  • Eric Weisstein's World of Mathematics, Sexy Primes. [The definition in this webpage is unsatisfactory, because it defines a "sexy prime" as a pair of primes.- _N. J. A. Sloane_, Mar 07 2021].

Crossrefs

Cf. A023201, A046117.
Cf. A023271, A046122, A046124.

Programs

  • Mathematica
    lst={};Do[p=Prime[n];If[PrimeQ[p+6]&&PrimeQ[p+12]&&PrimeQ[p+18], AppendTo[lst, p+12]], {n, 8!}];lst (* Vladimir Joseph Stephan Orlovsky, Aug 29 2008 *)
Select[Prime[Range[3000]],AllTrue[#+{-12,-6,6},PrimeQ]&] (* Harvey P. Dale, Jan 25 2023 *)

Formula

a(n) = A046122(n) + 6. - Amiram Eldar, Apr 22 2022

A046124 Last member of a sexy prime quadruple: value of p+18 such that p, p+6, p+12 and p+18 are all prime.

Original entry on oeis.org

23, 29, 59, 79, 269, 619, 659, 1109, 1499, 1619, 1759, 1879, 2389, 2689, 3319, 3929, 4019, 5119, 5399, 5449, 5659, 6329, 6379, 9479, 11839, 12119, 12659, 13469, 14639, 14759, 15809, 15919, 17489, 18229, 19489, 20359, 21499, 23339, 24109
Offset: 1

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Author

Eric W. Weisstein, Dec 11 1999

Keywords

Links

Crossrefs

Cf. A023271, A046122, A046123.

Programs

  • Magma
    [p+18: p in PrimesUpTo(30000) | IsPrime(p+6) and IsPrime(p+12) and IsPrime(p+18)]; // Vincenzo Librandi, Jan 07 2015
  • Mathematica
    lst={};Do[p=Prime[n];If[PrimeQ[p+6]&&PrimeQ[p+12]&&PrimeQ[p+18], AppendTo[lst, p+18]], {n, 8!}];lst (* Vladimir Joseph Stephan Orlovsky, Aug 29 2008 *)

Formula

a(n) = A023271(n)+18 = A046122(n)+12 = A046123(n)+6. - Michel Marcus, Jan 06 2015

A227282 First primes of arithmetic progressions of 7 primes each with the common difference 210.

Original entry on oeis.org

47, 179, 199, 409, 619, 829, 881, 1091, 1453, 3499, 3709, 3919, 10529, 10627, 10837, 10859, 11069, 11279, 14423, 20771, 22697, 30097, 30307, 31583, 31793, 32363, 41669, 75703, 93281, 95747, 120661, 120737, 120871, 120947, 129287, 140603, 153319, 153529
Offset: 1

Views

Author

Sameen Ahmed Khan, Jul 05 2013

Keywords

Comments

The minimal possible difference in an AP-k is conjectured to be k# for all k > 7.
For k = 7, we have d = 5*5# = 150 and there is ONLY one AP-7 with this difference: {7, 157, 307, 457, 607, 757, 907}.

Examples

			p = 179 then the AP-5 is {179, 389, 599, 809, 1019, 1229, 1439} with the difference 7# = 210.

Links

Crossrefs

Cf. A001359, A023241, A023271, A094220, A156204, A227281, A227283, A227284, A227285, A227286.

Programs

  • Mathematica
    Clear[p]; d = 210; ap7p = {}; Do[If[PrimeQ[{p, p + d, p + 2*d, p + 3*d, p + 4*d, p + 5*d, p + 6*d}] == {True, True, True, True, True, True, True}, AppendTo[ap7p, p]], {p, 3, 10^9, 2}]; ap7p
Select[Prime[Range[15000]],And@@PrimeQ[NestList[210+#&,#,6]]&] (* Harvey P. Dale, Nov 16 2013 *)
  • PARI
    is(p)=forstep(k=p,p+1260,210,if(!isprime(k),return(0)));1 \\ Charles R Greathouse IV, Dec 19 2013

A227283 First primes of arithmetic progressions of 8 primes each with the common difference 210.

Original entry on oeis.org

199, 409, 619, 881, 3499, 3709, 10627, 10859, 11069, 30097, 31583, 120661, 120737, 153319, 182537, 471089, 487391, 564973, 565183, 825991, 1010747, 1280623, 1288607, 1288817, 1302281, 1302491, 1395209, 1982599, 2358841, 2359051, 2439571, 3161017, 3600521
Offset: 1

Views

Author

Sameen Ahmed Khan, Jul 05 2013

Keywords

Comments

The minimal possible difference in an AP-k is conjectured to be k# for all k > 7.

Links

Crossrefs

Cf. A001359, A023241, A023271, A094220, A156204, A227281, A227282, A227284, A227285, A227286.

Programs

  • Mathematica
    Clear[p]; d = 210; ap8p = {}; Do[If[PrimeQ[{p, p + d, p + 2*d, p + 3*d, p + 4*d, p + 5*d, p + 6*d, p + 7*d}] == {True, True, True, True, True, True, True, True}, AppendTo[ap8p, p]], {p, 3, 3000000, 2}]; ap8p
Select[Prime[Range[260000]],AllTrue[NestList[#+210&,#,7],PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, May 03 2018 *)

A227285 First primes of arithmetic progressions of 11 primes each with the common difference 2310.

Original entry on oeis.org

60858179, 186874511, 291297353, 1445838451, 2943023729, 4597225889, 7024895393, 8620560607, 8656181357, 19033631401, 20711172773, 25366690189, 27187846201, 32022299977, 34351919351
Offset: 1

Views

Author

Sameen Ahmed Khan, Jul 05 2013

Keywords

Comments

The minimal possible difference in an AP-k is conjectured to be k# for all k > 7.
16th term is greater than 40*10^9.

Examples

			p = 186874511 then the AP-11 is {186874511, 186876821, 186879131, 186881441, 186883751, 186886061, 186888371, 186890681, 186892991, 186895301, 186897611} with the difference 11# = 2*3*5*7*11 = 2310.

Links

Crossrefs

Cf. A001359, A023241, A023271, A094220, A156204, A227281, A227282, A227283, A227284, A227286.

Programs

  • Mathematica
    Clear[p]; d = 2310; ap11p = {}; Do[If[PrimeQ[{p, p + d, p + 2*d, p + 3*d, p + 4*d, p + 5*d, p + 6*d, p + 7*d, p + 8*d, p + 9*d, p + 10*d}] == {True, True, True, True, True, True, True, True, True, True, True}, AppendTo[ap11p, p]], {p, 3, 40*10^9, 2}]; ap11p
ap11Q[n_]:=AllTrue[Rest[NestList[2310+#&,n,10]],PrimeQ]; Select[Prime[ Range[ 148*10^7]],ap11Q] (* The program uses the AllTrue function from Mathematica version 10 *) (* The program will take a long time to run *) (* Harvey P. Dale, Oct 27 2019 *)

Extensions

a(16)-a(21) from Zak Seidov, Jul 07 2014
Showing 1-10 of 22 results. Next

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

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