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

%I A014561 #72 Feb 16 2025 08:32:33
%S A014561 0,3,6,27,49,62,69,108,115,188,314,433,521,524,535,601,630,647,700,
%T A014561 742,843,1057,1161,1459,1711,1844,2099,2240,2316,2407,2575,2656,2701,
%U A014561 2757,2960,3261,3304,3370,3661,3884,3976,4073,4515,4805,5242,5523,5561,5705
%N A014561 Numbers k giving rise to prime quadruples (30k+11, 30k+13, 30k+17, 30k+19).
%C A014561 Intersection of A089160 and A089161. - _Zak Seidov_, Dec 22 2006
%C A014561 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
%C A014561 Comment from _Frank Ellermann_, Mar 13 2020: (Start)
%C A014561 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.
%C A014561 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:
%C A014561   13 * 17 = 210 + 11 for a(k) <> 7 + 1001 * k,
%C A014561   11 * 29 = 300 + 19 for a(k) <> 10 + 77 * k,
%C A014561   11 * 37 = 390 + 17 for a(k) <> 13 + 77 * k,
%C A014561   19 * 23 = 420 + 17 for a(k) <> 14 + 321321 * k,
%C A014561   17 * 31 = 510 + 17 for a(k) <> 17 + 17017 * k,
%C A014561   13 * 47 = 600 + 11 for a(k) <> 20 + 1001 * k,
%C A014561   11 * 59 = 630 + 19 for a(k) <> 21 + 77 * k, and
%C A014561   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}.
%C A014561 (End)
%H A014561 Michael De Vlieger, <a href="/A014561/b014561.txt">Table of n, a(n) for n = 1..10972</a> (first 1000 terms from Zak Seidov)
%H A014561 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimeQuadruplet.html">Prime Quadruplet</a>.
%F A014561 a(n) = (A007811(n) - 1)/3. - _Zak Seidov_, Sep 21 2009
%F A014561 a(n) = (A007530(n+1) - 11)/30 = floor(A007530(n+1)/30). - _M. F. Hasler_, Dec 05 2013
%F A014561 a(n) = A061668(n) - 1. - _Hugo Pfoertner_, Nov 03 2023
%e A014561 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).
%t A014561 a014561Q[n_Integer] :=
%t A014561   If[And[PrimeQ[30 n + 11], PrimeQ[30 n + 13], PrimeQ[30 n + 17],
%t A014561      PrimeQ[30 n + 19]] == True, True, False];
%t A014561 a014561[n_Integer] :=
%t A014561   Flatten[Position[Thread[a014561Q[Range[n]]], True]];
%t A014561 a014561[1000] (* _Michael De Vlieger_, Jul 17 2014 *)
%t A014561 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 *)
%o A014561 (PARI) isok(n) = isprime(30*n+11) && isprime(30*n+13) && isprime(30*n+17) && isprime(30*n+19) \\ _Michel Marcus_, Jun 09 2013
%Y A014561 Cf. A089160, A089161.
%Y A014561 Cf. A007530, A007811, A061668.
%Y A014561 A100418 and A100423 are subsequences.
%K A014561 easy,nonn
%O A014561 1,2
%A A014561 _Eric W. Weisstein_
%E A014561 More terms from _Warut Roonguthai_