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.

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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

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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.

A100418 Numbers k such that 30*k + {1,11,13,17,19,23,29} are all prime.

Original entry on oeis.org

49, 34083, 41545, 48713, 140609, 524027, 616812, 855281, 1314397, 1324750, 1636152, 2281293, 2927134, 3401412, 3605413, 4989341, 5212221, 5284979, 5406303, 5645269, 6141254, 6342728, 7231434, 7347697, 7637329, 8027068, 8161657, 8372756, 8392776, 8567216, 8986096, 9145563
Offset: 1

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Author

Ferenc Adorjan (fadorjan(AT)freemail.hu), Nov 19 2004

Keywords

Comments

Values are 0 mod 7.
From Peter Munn, Sep 06 2023: (Start)
In each case, the 7 primes are necessarily consecutive.
As A065706 demonstrates, many intervals of 27 integers contain 8 primes, but only A364678(30) = 7 primes can occur between adjacent positive multiples of 30. This is because there are 8 values {1,7,11,13,17,19,23,29} coprime to 30, but they cover every residue class modulo 7, which means at least one of 30*k + {1,7,11,13,17,19,23,29} is divisible by 7.
1 and 29 are in the same residue class, but if we remove any of the other coprime integers there is a class that is not represented in the set. For this sequence, we remove 7, so when k is a multiple of 7, none of 30*k + {1,11,13,17,19,23,29} is a multiple of 2, 3, 5 or 7 and the set can potentially be 7 consecutive primes.
The sequences for the other appropriate subsets of 7 coprime values are A100419-A100423.
(End)

Links

Crossrefs

Cf. A005776, A007775, A056956, A065706, A076205, A100419-A100423, A364678.

Programs

  • Magma
    [ n: n in [0..70000000 by 7] | forall{ q: q in [1, 11, 13, 17, 19, 23, 29] | IsPrime(30*n+q) } ]; // Klaus Brockhaus, Feb 24 2011
  • Mathematica
    Select[Range[803*10^4],AllTrue[30#+{1,11,13,17,19,23,29},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jun 11 2019 *)
  • PARI
    {pav7(mx)= local(wp=[1,11,13,17,19,23,29],v=[],i,j,m); for(k=1,mx, i=k*30;j=1;m=1;while(m&&(j<8),m=isprime(i+wp[j]);j+=1);if(m,v=concat(v,k))); return(v)}

Extensions

Edited by Don Reble, Nov 17 2005

A083409 Number of prime k-tuplet constellations, i.e., patterns with minimal diameter A008407.

Original entry on oeis.org

1, 2, 1, 2, 1, 2, 3, 4, 2, 2, 2, 6, 2, 4, 2, 4, 2, 4, 2, 2, 4, 2, 4, 18, 2, 8, 10, 2, 2, 2, 4, 14, 20, 2, 2, 2, 6, 26, 26, 8, 2, 6, 18, 4, 4, 4, 2, 2, 22, 22, 2, 2, 26, 6, 6, 2, 2, 4, 2, 2, 6, 2, 2, 2, 2, 18, 2, 20, 2, 2, 2, 10, 2, 14, 14, 40, 8, 2, 14, 14, 16, 4, 2, 2, 60, 50, 2, 2, 2, 16, 2, 18, 12
Offset: 2

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Author

Frank Ellermann, Jun 07 2003

Keywords

Examples

			For a(8) = 3 octuplet patterns see A065706. for a(6) = 1 sextet see A061671.

Links

Crossrefs

Cf. A008407, A020497, A023193, A061671, A065688, A065706.

Extensions

More terms from Engelsma's website sent by T. D. Noe, Jul 21 2006

A375344 First term p1 of octuplets of consecutive prime numbers pi with given successive gaps pi-p1, i=2, ...,8 (6, 8, 18, 24, 30, 36, 38).

Original entry on oeis.org

233, 2721413, 154670903, 200559053, 232777673, 273788363, 299267663, 459117353, 527326403, 1015923113, 1563572243, 1688692763, 2426018723, 2918492243, 3743134523, 4445599853, 4458163943, 4697619593, 5493835013, 5546977823, 5930389313, 6131660663, 6470661143, 7598587943
Offset: 1

Views

Author

René-Louis Clerc, Aug 12 2024

Keywords

Comments

The choice of successive gaps (6, 8, 18, 24, 30, 36, 38) is such that the sum of the eight prime numbers beginning with 233 is 2024. The next year being the sum of analogous octuplet is 21771464 (21772nd millenium).

Examples

			233, 239, 241, 251, 257, 263, 269, 271 (sum = 2024).
2721413, 2721419, 2721421, 2721431, 2721437, 2721443, 2721449, 2721451 (sum = 21771464).

Links

Crossrefs

Cf. A000040, A001223, A002476, A151800, A000720, A065706, A022011.

Programs

  • PARI
    uplet(p)= {n=0;for(i=p, p+38, if(isprime(i), n+=1));n}
octo(m)={for(p=3,p=10^m,if(isprime(p) && isprime(p+6) && isprime(p+8) && isprime(p+18) && isprime(p+24) && isprime(p+30) && isprime(p+36) && isprime(p+38) && uplet(p)==8,print1(p,", ")))}
listocto(p1)=print1(p1,", ", p1+6,", ", p1+8,", ", p1+18,", ", p1+24,", ", p1+30,", ", p1+36", ", p1+38)
Previous Showing 31-34 of 34 results.

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