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

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

1,2

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)

Links

Michael De Vlieger, Table of n, a(n) for n = 1..10972 (first 1000 terms from Zak Seidov)

Eric Weisstein's World of Mathematics, Prime Quadruplet.

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

Example

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).

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 *)

Prog

(PARI) isok(n) = isprime(30*n+11) && isprime(30*n+13) && isprime(30*n+17) && isprime(30*n+19) \\ Michel Marcus, Jun 09 2013

Crossrefs

Cf. A089160, A089161.

Cf. A007530, A007811, A061668.

A100418 and A100423 are subsequences.

Sequence in context: A256762 A064283 A266857 * A034502 A217725 A023169

Adjacent sequences: A014558 A014559 A014560 * A014562 A014563 A014564

Keyword

easy,nonn

Author

Eric W. Weisstein

Extensions

More terms from Warut Roonguthai

Status

approved