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A350543 Twin primes x, represented by their average, such that x is the first and x+48 the last of five successive twins.

%I A350543 #13 Mar 04 2023 11:32:34
%S A350543 12,3919212,325267932,905119332,2013256362,3066212112,3240097962,
%T A350543 4046054430,6567515370,7561533402,10816172202,10895874132,17444777880,
%U A350543 20905115040,22194295812,23641113912,26079344100,26368755222,27350615220,29861090682,33240296052
%N A350543 Twin primes x, represented by their average, such that x is the first and x+48 the last of five successive twins.
%C A350543 Subsequence of A014574. The terms represent quintuples of twin primes. As there are only 31 terms < 10^11, the accordance with the k-tuple conjecture is not very good, see links "k-tuple conjecture" and A350541, "Test of the k-tuple conjecture". Moreover, the formalism of the conjecture allows the evaluation of the expected frequencies of eight types of quintuples relative to the frequency of all quintuples. The differences are considerable:
%C A350543              relative frequencies
%C A350543 Example    observed     expected
%C A350543    (1)     11/31=35.5%    23.7%
%C A350543    (2)      5/31=16.1%    15.0%
%C A350543    (3)      3/31= 9.7%     7.5%
%C A350543    (4)      6/31=19.4%    23.7%
%C A350543    (5)      3/31= 9.7%    15.0%
%C A350543    (6)      2/31= 6.5%     3.8%
%C A350543    (7)      1/31= 3.2%     7.5%
%C A350543    (8)          0          3.8%
%C A350543 Generalization:
%C A350543 Twin primes x such that x is the first and x+d the last of m successive twins.
%C A350543 m   d
%C A350543 1   0  A014574(n) twin primes
%C A350543 2   6  A173037(n)-3
%C A350543 3  12  Only one quadruple: (6,12,18,30)
%C A350543 3  18  A350541
%C A350543 4  24  Only one quintuple: (6,12,18,30,42)
%C A350543 4  30  A350542
%C A350543 5  36  6, 39713433660, 66419473020, 71525244600*
%C A350543 5  42  18873492, 180929682, 1170073332, 2550576612, 5807487204, 27523454232, 33497368554, 50062053714, 63167632254, 86883508944, 99939276954*
%C A350543 5  48  Current sequence
%C A350543 Annotations:
%C A350543 *The number of terms < 10^11 is too small for submitting a new sequence.
%C A350543 (m=5,d=30) is empty for divisibility reasons.
%H A350543 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/k-TupleConjecture.html">k-Tuple Conjecture</a>.
%e A350543 The quintuples of twins have the form (x,x+a,x+b,x+c,x+d)
%e A350543   (a,b,c,d)      least example
%e A350543 (1)  6,18,30,48  x=           12
%e A350543 (2)  6,30,36,48  x=    123919212
%e A350543 (3)  6,18,36,48  x= 123240097962
%e A350543 (4) 18,30,42,48  x= 124046054430
%e A350543 (5) 12,18,42,48  x=1217444777880
%e A350543 (6) 12,18,30,48  x=1220905115040
%e A350543 (7) 12,30,42,48  x=1227350615220
%e A350543 (8) 18,30,42,48  x>10^11
%o A350543 (Maxima)
%o A350543 block(twin:[6], n:1,  p1:11,  j2:1, nmax: 3,
%o A350543 /*returns nmax terms*/
%o A350543 m:5, d:48, w: makelist(-d, i, 1, m),
%o A350543 while n<nmax  do(
%o A350543 p2: next_prime(p1), if p2-p1=2 then(
%o A350543   k:p1+1, j1:j2, j2:1+ mod(j2,m), w[j1]:k,
%o A350543   if w[j1]-w[j2]=d then(n:n+1, twin: append(twin,[k-d]))),
%o A350543 p1:p2), twin);
%Y A350543 Cf. A014574, A173037, A350541, A350542.
%K A350543 nonn
%O A350543 1,1
%A A350543 _Gerhard Kirchner_, Jan 07 2022