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A350541 Twin primes x, represented by their average, such that x is the first and x+18 the last of three successive twins.

%I A350541 #20 Feb 28 2023 13:07:28
%S A350541 12,180,810,5640,9420,18042,62970,88800,97842,109830,165702,284730,
%T A350541 392262,452520,626610,663570,663582,855720,983430,1002342,1003350,
%U A350541 1068702,1146780,1155612,1322160,1329702,1592862,1678752,1718862,1748472,2116560,2144490
%N A350541 Twin primes x, represented by their average, such that x is the first and x+18 the last of three successive twins.
%C A350541 Subsequence of A014574. For x>6, d=18 is the least possible difference between the least and the greatest of three twins. For d=12, one of the six terms 6*k+-1, 6*k+6+-1,6*k+12+-1 would be divisible by 5. Therefore, d>12, except for x=6.
%C A350541 The distribution of 35314 terms < 10^11 is in accordance with the k-tuple conjecture, see links "k-tuple conjecture" and "Test of the k-tuple conjecture".
%C A350541 Generalizations:
%C A350541 Twin primes x such that x is the first and x+d the last of m successive twins.
%C A350541 m   d
%C A350541 1   0  A014574(n) twin primes
%C A350541 2   6  A173037(n)-3
%C A350541 3  12  Only one quadruple: (6,12,18,30)
%C A350541 3  18  Current sequence
%C A350541 4  24  Only one quintuple: (6,12,18,30,42)
%C A350541 4  30  A350542
%C A350541 5  36  See  A350543
%C A350541 5  42  See  A350543
%C A350541 5  48  A350543
%H A350541 Gerhard Kirchner, <a href="/A350541/a350541.pdf">Test of the k-tuple conjecture</a>
%H A350541 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/k-TupleConjecture.html">k-tuple conjecture</a>.
%e A350541 Triples of twins Example   6-tuple of primes
%e A350541 (x,x+ 6,x+18)     x= 12   (11,13,17,19,29,31)
%e A350541 (x,x+12,x+18)     x=180   (179,181,191,193,197,199)
%t A350541 Select[Prime@Range[4,160000],Count[Range[#,#+18],_?(PrimeQ@#&&PrimeQ[#+2]&)]==3&]+1 (* _Giorgos Kalogeropoulos_, Jan 07 2022 *)
%o A350541 (Maxima)
%o A350541 block(twin:[], n:0, p1:11, j2:1, nmax: 3,
%o A350541 /*returns nmax terms*/
%o A350541 m:3, d:18, w: makelist(-d,i,1,m),
%o A350541 while n<nmax  do(
%o A350541 p2: next_prime(p1), if p2-p1=2 then(
%o A350541   k:p1+1, j1:j2, j2:1+ mod(j2,m), w[j1]:k,
%o A350541   if w[j1]-w[j2]=d then(n:n+1, twin: append(twin,[k-d]))),
%o A350541 p1:p2), twin);
%Y A350541 Cf. A014574, A173037, A350542, A350543.
%K A350541 nonn
%O A350541 1,1
%A A350541 _Gerhard Kirchner_, Jan 06 2022