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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.

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A257373 Initial members of prime 17-tuplets.

Original entry on oeis.org

13, 17, 1620784518619319025971, 2639154464612254121531, 3259125690557440336631, 37630850994954402655487, 47624415490498763963983, 53947453971035573715707, 78314167738064529047713, 83405687980406998933663, 110885131130067570042703, 124211857692162527019731
Offset: 1

Views

Author

Tim Johannes Ohrtmann, Apr 21 2015

Keywords

Links

Crossrefs

Initial members of all of the first prime k-tuplets:
twin primes: A001359.
prime triples: A007529 out of A022004, A022005.
prime quadruplets: A007530.
prime 5-tuples: A086140 out of A022007, A022006.
prime sextuplets: A022008.
prime septuplets: A257124 out of A022009, A022010.
prime octuplets: A065706 out of A022011, A022012, A022013.
prime nonuplets: A257125 out of A022547, A022548, A022545, A022546.
prime decaplets: A257127 out of A027569, A027570.
prime 11-tuplets: A257129 out of A213646, A213647.
prime 12-tuplets: A257131 out of A213601, A213645.
prime 13-tuplets: A257135 out of A214947, A257137, A257138, A257139, A257140, A257141.
prime 14-tuplets: A257166 out of A257167, A257168.
prime 15-tuplets: A257169 out of A257304, A257305, A257306, A257307.
prime 16-tuplets: A257308 out of A257369, A257370.
prime 17-tuplets: this sequence out of A257374, A257375, A257376, A257377.

A257374 Numbers n such that n, n+4, n+10, n+12, n+16, n+22, n+24, n+30, n+36, n+40, n+42, n+46, n+52, n+54, n+60, n+64 and n+66 are all prime.

Original entry on oeis.org

734975534793324512717947, 753314125249587933791677, 1341829940444122313597407
Offset: 1

Views

Author

Tim Johannes Ohrtmann, Apr 21 2015

Keywords

Links

Crossrefs

Initial members of all of the first prime k-tuplets:
twin primes: A001359.
prime triples: A007529 out of A022004, A022005.
prime quadruplets: A007530.
prime 5-tuples: A086140 out of A022007, A022006.
prime sextuplets: A022008.
prime septuplets: A257124 out of A022009, A022010.
prime octuplets: A065706 out of A022011, A022012, A022013.
prime nonuplets: A257125 out of A022547, A022548, A022545, A022546.
prime decaplets: A257127 out of A027569, A027570.
prime 11-tuplets: A257129 out of A213646, A213647.
prime 12-tuplets: A257131 out of A213601, A213645.
prime 13-tuplets: A257135 out of A214947, A257137, A257138, A257139, A257140, A257141.
prime 14-tuplets: A257166 out of A257167, A257168.
prime 15-tuplets: A257169 out of A257304, A257305, A257306, A257307.
prime 16-tuplets: A257308 out of A257369, A257370.
prime 17-tuplets: A257373 out of this sequence, A257375, A257376, A257377.

Extensions

a(3) from Norman Luhn, Oct 27 2021

A257375 Numbers n such that n, n+4, n+6, n+10, n+16, n+18, n+24, n+28, n+30, n+34, n+40, n+46, n+48, n+54, n+58, n+60 and n+66 are all prime.

Original entry on oeis.org

13, 47624415490498763963983, 78314167738064529047713, 83405687980406998933663, 110885131130067570042703, 163027495131423420474913
Offset: 1

Views

Author

Tim Johannes Ohrtmann, Apr 21 2015

Keywords

Links

Crossrefs

Initial members of all of the first prime k-tuplets:
twin primes: A001359.
prime triples: A007529 out of A022004, A022005.
prime quadruplets: A007530.
prime 5-tuples: A086140 out of A022007, A022006.
prime sextuplets: A022008.
prime septuplets: A257124 out of A022009, A022010.
prime octuplets: A065706 out of A022011, A022012, A022013.
prime nonuplets: A257125 out of A022547, A022548, A022545, A022546.
prime decaplets: A257127 out of A027569, A027570.
prime 11-tuplets: A257129 out of A213646, A213647.
prime 12-tuplets: A257131 out of A213601, A213645.
prime 13-tuplets: A257135 out of A214947, A257137, A257138, A257139, A257140, A257141.
prime 14-tuplets: A257166 out of A257167, A257168.
prime 15-tuplets: A257169 out of A257304, A257305, A257306, A257307.
prime 16-tuplets: A257308 out of A257369, A257370.
prime 17-tuplets: A257373 out of A257374, this sequence, A257376, A257377.

A257376 Numbers n such that n, n+6, n+8, n+12, n+18, n+20, n+26, n+32, n+36, n+38, n+42, n+48, n+50, n+56, n+60, n+62 and n+66 are all prime.

Original entry on oeis.org

1620784518619319025971, 2639154464612254121531, 3259125690557440336631, 124211857692162527019731
Offset: 1

Views

Author

Tim Johannes Ohrtmann, Apr 21 2015

Keywords

Links

Crossrefs

Initial members of all of the first prime k-tuplets:
twin primes: A001359.
prime triples: A007529 out of A022004, A022005.
prime quadruplets: A007530.
prime 5-tuples: A086140 out of A022007, A022006.
prime sextuplets: A022008.
prime septuplets: A257124 out of A022009, A022010.
prime octuplets: A065706 out of A022011, A022012, A022013.
prime nonuplets: A257125 out of A022547, A022548, A022545, A022546.
prime decaplets: A257127 out of A027569, A027570.
prime 11-tuplets: A257129 out of A213646, A213647.
prime 12-tuplets: A257131 out of A213601, A213645.
prime 13-tuplets: A257135 out of A214947, A257137, A257138, A257139, A257140, A257141.
prime 14-tuplets: A257166 out of A257167, A257168.
prime 15-tuplets: A257169 out of A257304, A257305, A257306, A257307.
prime 16-tuplets: A257308 out of A257369, A257370.
prime 17-tuplets: A257373 out of A257374, A257375, this sequence, A257377.

Extensions

a(1) corrected by Tim Johannes Ohrtmann, Dec 17 2015

A257377 Numbers n such that n, n+2, n+6, n+12, n+14, n+20, n+24, n+26, n+30, n+36, n+42, n+44, n+50, n+54, n+56, n+62 and n+66 are all prime.

Original entry on oeis.org

17, 37630850994954402655487, 53947453971035573715707, 174856263959258260646207, 176964638100452596444067, 207068890313310815346497, 247620555224812786876877, 322237784423505559739147
Offset: 1

Views

Author

Tim Johannes Ohrtmann, Apr 21 2015

Keywords

Links

Crossrefs

Initial members of all of the first prime k-tuplets:
twin primes: A001359.
prime triples: A007529 out of A022004, A022005.
prime quadruplets: A007530.
prime 5-tuples: A086140 out of A022007, A022006.
prime sextuplets: A022008.
prime septuplets: A257124 out of A022009, A022010.
prime octuplets: A065706 out of A022011, A022012, A022013.
prime nonuplets: A257125 out of A022547, A022548, A022545, A022546.
prime decaplets: A257127 out of A027569, A027570.
prime 11-tuplets: A257129 out of A213646, A213647.
prime 12-tuplets: A257131 out of A213601, A213645.
prime 13-tuplets: A257135 out of A214947, A257137, A257138, A257139, A257140, A257141.
prime 14-tuplets: A257166 out of A257167, A257168.
prime 15-tuplets: A257169 out of A257304, A257305, A257306, A257307.
prime 16-tuplets: A257308 out of A257369, A257370.
prime 17-tuplets: A257373 out of A257374, A257375, A257376, this sequence.

A014561 Numbers k giving rise to prime quadruples (30k+11, 30k+13, 30k+17, 30k+19).

Original entry on oeis.org

0, 3, 6, 27, 49, 62, 69, 108, 115, 188, 314, 433, 521, 524, 535, 601, 630, 647, 700, 742, 843, 1057, 1161, 1459, 1711, 1844, 2099, 2240, 2316, 2407, 2575, 2656, 2701, 2757, 2960, 3261, 3304, 3370, 3661, 3884, 3976, 4073, 4515, 4805, 5242, 5523, 5561, 5705
Offset: 1

Views

Author

Eric W. Weisstein

Keywords

Comments

Intersection of A089160 and A089161. - Zak Seidov, Dec 22 2006
This can be seen as a condensed version of A007530, which lists the first member of the actual prime quadruplet (30x+11, 30x+13, 30x+17, 30x+19), x=a(n). - M. F. Hasler, Dec 05 2013
Comment from Frank Ellermann, Mar 13 2020: (Start)
Ignoring 2 and 3, {5,7,11,13} is the only twin-twin prime quadruple not following this pattern for primes > 5. One candidate mod 30 corresponds to 7 candidates mod 210, but 7 * 7 = 30 + 19, 7 * 11 = 60 + 17, 7 * 19 = 120 + 13, and 7 * 23 = 190 + 11 are multiples of 7, leaving only 3 candidates mod 210.
Likewise, 13 * 13 = 150 + 19 is a multiple of 13 mod 30030, but 5 + 1001 * k is a proper subset of 5 + 7 * k with 1001 = 13 * 11 * 7. Other disqualified candidates with nonzero k are:
13 * 17 = 210 + 11 for a(k) <> 7 + 1001 * k,
11 * 29 = 300 + 19 for a(k) <> 10 + 77 * k,
11 * 37 = 390 + 17 for a(k) <> 13 + 77 * k,
19 * 23 = 420 + 17 for a(k) <> 14 + 321321 * k,
17 * 31 = 510 + 17 for a(k) <> 17 + 17017 * k,
13 * 47 = 600 + 11 for a(k) <> 20 + 1001 * k,
11 * 59 = 630 + 19 for a(k) <> 21 + 77 * k, and
11 * 67 = 720 + 17 for a(k) <> 24 + 77 + k, picking the smallest prime factors 11, 17, 11 for {407, 527, 737} instead of 13, 23, 17 for {403, 529, 731}.
(End)

Examples

			a(4) = 27 for 27*30 = 810 yields twin primes at 810+11 = A001359(32) = A000040(142) and 810+17 = A001359(33) = A000040(144) ending at 810+19 = A000040(145).

Links

Crossrefs

Cf. A089160, A089161.
Cf. A007530, A007811, A061668.
A100418 and A100423 are subsequences.

Programs

  • Mathematica
    a014561Q[n_Integer] :=
  If[And[PrimeQ[30 n + 11], PrimeQ[30 n + 13], PrimeQ[30 n + 17],
     PrimeQ[30 n + 19]] == True, True, False];
a014561[n_Integer] :=
  Flatten[Position[Thread[a014561Q[Range[n]]], True]];
a014561[1000] (* Michael De Vlieger, Jul 17 2014 *)
Select[Range[0,6000],AllTrue[30#+{11,13,17,19},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Oct 21 2016 *)
  • PARI
    isok(n) = isprime(30*n+11) && isprime(30*n+13) && isprime(30*n+17) && isprime(30*n+19) \\ Michel Marcus, Jun 09 2013

Formula

a(n) = (A007811(n) - 1)/3. - Zak Seidov, Sep 21 2009
a(n) = (A007530(n+1) - 11)/30 = floor(A007530(n+1)/30). - M. F. Hasler, Dec 05 2013
a(n) = A061668(n) - 1. - Hugo Pfoertner, Nov 03 2023

Extensions

More terms from Warut Roonguthai

A078857 Initial term in sequence of four consecutive primes separated by 3 consecutive differences each <=6 (i.e., when d=2,4 or 6) and forming d-pattern=[6, 6,2]; short d-string notation of pattern = [662].

Original entry on oeis.org

47, 167, 257, 557, 587, 647, 1217, 2957, 4007, 6257, 6857, 7577, 10847, 11927, 14537, 16217, 17477, 19457, 24407, 25457, 26687, 26717, 29867, 41507, 41597, 48527, 51407, 54617, 56087, 60077, 61547, 68477, 75527, 82457, 84047, 94427, 101267
Offset: 1

Views

Author

Labos Elemer, Dec 11 2002

Keywords

Comments

Subsequence of A047948. - R. J. Mathar, Feb 11 2013

Examples

			p=47,47+6=53,47+6+6=59,47+6+6+2=61 are consecutive primes.

Links

Crossrefs

Cf. analogous prime quadruple sequences with various possible {2, 4, 6}-difference-patterns in brackets: A007530[242], A078847[246], A078848[264], A078849[266], A052378[424], A078850[426], A078851[462], A078852[466], A078853[624], A078854[626], A078855[642], A078856[646], A078857[662], A078858[664], A033451[666].

Programs

  • Mathematica
    Select[Partition[Prime[Range[10000]],4,1],Differences[#]=={6,6,2}&][[All,1]] (* Harvey P. Dale, Apr 29 2017 *)

Formula

Primes p = p(i) such that p(i+1)=p+6, p(i+2)=p+6+6, p(i+3)=p+6+6+2.

Extensions

Listed terms verified by Ray Chandler, Apr 20 2009

A078858 Initial term in sequence of four consecutive primes separated by 3 consecutive differences each <=6 (i.e., when d = 2, 4 or 6) and forming d-pattern = [6, 6, 4]; short d-string notation of pattern = [664].

Original entry on oeis.org

151, 367, 601, 727, 2281, 2671, 3307, 4987, 5557, 10651, 12967, 13171, 15907, 18217, 18427, 20101, 20341, 24091, 27061, 28591, 30097, 30307, 31321, 32491, 35311, 37951, 41941, 42181, 42391, 45751, 52951, 53617, 55201, 56767, 59107, 65407
Offset: 1

Views

Author

Labos Elemer, Dec 11 2002

Keywords

Comments

Subsequence of A047948. - R. J. Mathar, Feb 11 2013

Examples

			p=151, 151+6 = 157, 151+6+6 = 163, 151+6+6+4 = 167 are consecutive primes.

Links

Crossrefs

Cf. analogous prime quadruple sequences with various possible {2, 4, 6}-difference-patterns in brackets: A007530[242], A078847[246], A078848[264], A078849[266], A052378[424], A078850[426], A078851[462], A078852[466], A078853[624], A078854[626], A078855[642], A078856[646], A078857[662], A078858[664], A033451[666].

Programs

  • Mathematica
    Transpose[Select[Partition[Prime[Range[6600]],4,1],Differences[#] == {6,6,4}&]][[1]] (* Harvey P. Dale, Nov 04 2011 *)

Formula

Primes p = p(i) such that p(i+1) = p+6, p(i+2) = p+6+6, p(i+3) = p+6+6+4.

Extensions

Listed terms verified by Ray Chandler, Apr 20 2009

A031165 a(n) = prime(n+3) - prime(n).

Original entry on oeis.org

5, 8, 8, 10, 8, 10, 12, 12, 14, 12, 12, 10, 12, 16, 14, 14, 12, 12, 12, 12, 16, 18, 18, 14, 10, 8, 10, 20, 22, 24, 12, 18, 14, 18, 14, 16, 16, 16, 14, 18, 14, 16, 8, 18, 26, 28, 18, 10, 12, 12, 18, 18, 22, 18, 14, 14, 12, 12, 16, 26, 28, 20, 10, 20, 24, 30, 18, 16, 12
Offset: 1

Views

Author

Jeff Burch, Dec 11 1999

Keywords

Comments

Comments from Jonathan Vos Post, Jan 22 2006 (Start): This sequence is the k=3 case of the family of sequences a(k,n) = prime(n+k) - prime(n). See A001223 and A031131 for k = 1 and 2.
The records in this sequence give A115401. The minimal value, after the anomalous initial values (5, 8, 8), is 8 which occurs iff n is an element of A007530 (prime quadruples: numbers n such that n, n+2, n+6, n+8 are all prime). (End)

Examples

			a(1) = prime(4) - prime(1) = 7 - 2 = 5, which is the only odd element of this sequence.
a(2) = prime(5) - prime(2) = 11 - 3 = 8.
a(3) = prime(6) - prime(3) = 13 - 5 = 8.
a(4) = prime(7) - prime(4) = 17 - 7 = 10.
a(99) = prime(102) - prime(99) = 557 - 523 = 34. - _Jonathan Vos Post_, Jan 22 2006

Links

Crossrefs

Cf. A000040, A001223, A007530, A031131, A034961, A115401.
Cf. A031166, A031167, A031168, A031169, A031170, A031171, A031172.

Programs

  • Haskell
    a031165 n = a031165_list !! (n-1)
a031165_list = zipWith (-) (drop 3 a000040_list) a000040_list
-- Reinhard Zumkeller, Aug 23 2015
  • Magma
    [NthPrime(n+3)-NthPrime(n): n in [1..100] ]; // Vincenzo Librandi, Apr 11 2011
  • Maple
    a:= n-> ithprime(n+3)-ithprime(n): seq (a(n), n=1..80);
  • Mathematica
    t = Array[Prime, 75]; Drop[t, 3] - Drop[t, -3] (* Robert G. Wilson v *)
#[[4]]-#[[1]]&/@Partition[Prime[Range[80]],4,1] (* Harvey P. Dale, Nov 07 2021 *)
  • PARI
    p=2;q=3;r=5;forprime(s=7,1e3,print1(s-p", "); p=q;q=r;r=s) \\ Charles R Greathouse IV, Nov 07 2012

Formula

a(n) = prime(n+3) - prime(n). a(n) = A000040(n+3) - A000040(n). - Jonathan Vos Post, Jan 22 2006
a(n) = A034961(n+1) - A034961(n). - Zak Seidov, Nov 07 2012

Extensions

Edited by R. J. Mathar and N. J. A. Sloane, Aug 11 2008

A078854 Initial term in sequence of four consecutive primes separated by 3 consecutive differences each <=6 (i.e., when d=2,4 or 6) and forming d-pattern=[6, 2,6]; short d-string notation of pattern = [626].

Original entry on oeis.org

23, 53, 263, 563, 593, 1223, 1283, 1613, 2333, 2543, 3533, 4013, 4643, 5843, 6263, 6353, 6563, 10853, 11483, 14543, 15263, 17483, 19073, 19373, 19463, 23663, 26723, 29123, 32363, 34253, 41603, 48473, 49193, 49523, 51413, 51473, 71333, 75983
Offset: 1

Views

Author

Labos Elemer, Dec 11 2002

Keywords

Comments

Subsequence of A049438. - R. J. Mathar, May 06 2017

Examples

			p=23,23+6=29,23+6+2=31,23+6+2+6=37 are consecutive primes.

Links

Crossrefs

Cf. analogous prime quadruple sequences with various possible {2, 4, 6}-difference-patterns in brackets: A007530[242], A078847[246], A078848[264], A078849[266], A052378[424], A078850[426], A078851[462], A078852[466], A078853[624], A078854[626], A078855[642], A078856[646], A078857[662], A078858[664], A033451[666].

Programs

  • Mathematica
    Transpose[Select[Partition[Prime[Range[7500]],4,1],Differences[#]=={6,2,6}&]][[1]] (* Harvey P. Dale, Apr 17 2015 *)

Formula

Primes p = p(i) such that p(i+1)=p+6, p(i+2)=p+6+2, p(i+3)=p+6+2+6.

Extensions

Listed terms verified by Ray Chandler, Apr 20 2009
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