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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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

A331840 Numbers k such that 30*k-13, 30*k-11 are twin primes.

Original entry on oeis.org

1, 4, 5, 7, 8, 12, 21, 28, 29, 43, 48, 50, 54, 56, 57, 60, 63, 67, 68, 70, 75, 76, 89, 90, 106, 109, 116, 118, 119, 126, 131, 138, 139, 141, 145, 151, 152, 155, 160, 166, 181, 183, 189, 196, 207, 228, 232, 238, 244, 249, 250, 252, 259, 263, 270, 280, 285, 287
Offset: 1

Views

Author

Frank Ellermann, Feb 26 2020

Keywords

Comments

All twin primes > 7 have the form 30*k-{13,11}, or 30*k +-1 (A176114), or 30*k+{11,13} (A089160).
All twin primes > 7 with least significant decimal digit 7 have the form 30*k-13.
All twin primes > 7 with least significant decimal digit 3 have the form 30*k+13.

Examples

			1 is a term because 1*30 - 13 =  17 = prime(6)  and 1*30 - 11 =  19 = prime(7).
4 is a term because 4*30 - 13 = 107 = prime(28) and 4*30 - 11 = 109 = prime(29).
5 is a term because 5*30 - 13 = 137 = prime(33) and 5*30 - 11 = 139 = prime(34).

Crossrefs

Cf. A089160, A089161, A176114, A332772.
Cf. A000040, A001097, A002822, A132242, A282323, A282324.

Programs

  • Mathematica
    Select[Range[300], And @@ PrimeQ[30*# - {11, 13}] &] (* Amiram Eldar, Feb 29 2020 *)
  • PARI
    isok(k) = isprime(30*k-13) && isprime(30*k-11); \\ Michel Marcus, Feb 29 2020
  • Rexx
    S = 1
do N = 2 while length( S ) < 255
   if NOPRIME( N*30 -13 )  then  iterate N
   if NOPRIME( N*30 -11 )  then  iterate N
   S = S || ',' N
end N
say S

Formula

a(n) = A089161(n)+1.
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