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A350542 Twin primes x, represented by their average, such that x is the first and x+30 the last of four successive twins.

%I A350542 #17 Mar 04 2023 11:32:29
%S A350542 12,626598,663570,1322148,2144478,2668218,6510192,6937938,10187910,
%T A350542 11495580,11721768,18873498,18873510,25658430,39659532,39851292,
%U A350542 46533468,80572158,84099318,86944602,91814712,93956100,123911532,128469150,129902022,148979838
%N A350542 Twin primes x, represented by their average, such that x is the first and x+30 the last of four successive twins.
%C A350542 Subsequence of A014574. For x>6, d=30 is the least possible difference between the least and the greatest of four twins. With d=24, six primes would have the form 6*k+-1, 6*k+6+-1,6*k+12+-1 which is impossible because one of the six numbers would be divisible by 5. Therefore, d>24, except for x=6. The distribution of 1134 terms < 10^11 is in accordance with the k-tuple conjecture, see links "k-tuple conjecture" and A350541, "Test of the k-tuple conjecture".
%C A350542 Generalization:
%C A350542 Twin primes x such that x is the first and x+d the last of m successive twins.
%C A350542 m   d
%C A350542 1   0  A014574(n) twin primes
%C A350542 2   6  A173037(n)-3
%C A350542 3  12  Only one quadruple: (6,12,18,30)
%C A350542 3  18  A350541
%C A350542 4  24  Only one quintuple: (6,12,18,30,42)
%C A350542 4  30  Current sequence
%C A350542 5  36  See A350543
%C A350542 5  42  See A350543
%C A350542 5  48  A350543
%H A350542 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/k-TupleConjecture.html">k-Tuple Conjecture</a>.
%e A350542 Quadruples of twins  Example       8-tuple of primes
%e A350542 (x,x+ 6,x+18,x+30)   x=12      (11,13,17,19,29,31,41,43)
%e A350542 (x,x+12,x+24,x+30)   x=626598  (x-1,x+1,x+11,x+13,x+23,x+25,x+29,x+31)
%e A350542 (x,x+12,x+18,x+30)   x=663570  (x-1,x+1,x+11,x+13,x+17,x+19,x+29,x+31)
%e A350542 (x,x+ 6,x+24,x+30), (x,x+6,x+12,x+30) and (x,x+18,x+24,x+30) do not occur for divisibility reasons.
%t A350542 Select[Prime@Range[4,200000], Count[Range[#,#+30],_?(PrimeQ@#&&PrimeQ[#+2]&)]==4&]+1 (* _Giorgos Kalogeropoulos_, Jan 07 2022 *)
%o A350542 (Maxima)
%o A350542 block(twin:[], n:0, p1:11, j2:1, nmax: 3,
%o A350542 /*returns nmax terms*/
%o A350542 m:4, d:30, w: makelist(-d,i,1,m),
%o A350542 while n<nmax  do(
%o A350542 p2: next_prime(p1), if p2-p1=2 then(
%o A350542   k:p1+1, j1:j2, j2:1+ mod(j2,m), w[j1]:k,
%o A350542   if w[j1]-w[j2]=d then(n:n+1, twin: append(twin,[k-d]))),
%o A350542 p1:p2), twin);
%Y A350542 Cf. A014574, A173037, A350541, A350543.
%K A350542 nonn
%O A350542 1,1
%A A350542 _Gerhard Kirchner_, Jan 07 2022