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A202319 Lesser of two semiprimes sandwiched each between semiprimes thus forming a twin semiprime-triple.

%I A202319 #33 Feb 26 2022 21:20:53
%S A202319 214,143098,194758,206134,273418,684898,807658,1373938,1391758,
%T A202319 1516534,1591594,1610998,1774798,1882978,1891762,2046454,2051494,
%U A202319 2163418,2163958,2338054,2359978,2522518,2913838,3108202,4221754,4297318,4334938,4866118,4988878,5108794
%N A202319 Lesser of two semiprimes sandwiched each between semiprimes thus forming a twin semiprime-triple.
%C A202319 Conjecture: For every natural number k there are infinitely many semiprime pairs sp and sp' both sandwiched between semiprimes such that sp' - sp = 4k.
%C A202319 Note: For the case k=1 the pair of two consecutive semiprime triples will be called twin semiprime-triple with an analogy to twin prime.
%C A202319 The number centered between the triples of a twin semiprime-triple must be divisible by 36. Let m be the middle of such a twin semiprime-triple. It is trivial that m is divisible by 4, and that it is congruent to 0, 4, or 5 (mod 9). If it were congruent to 4, then m-1 and m+2 would both be divisible by 3, hence equal to 3 times a prime. But then those two primes would differ by 1, impossible except for primes 2, 3, which can be checked separately. A similar argument eliminates the case m == 5 (mod 9), so m must be divisible by 9. Conjecture by the author, proved by _Franklin T. Adams-Watters_, Dec 18 2011.
%C A202319 Members of this sequence must be twice the lesser of a twin prime. - _Franklin T. Adams-Watters_, Dec 18 2011
%C A202319 A number is in the sequence if and only if it has the form 6k-2, with 2k+-1 being twin primes, 3k+-1 twin primes, and 6k+-1 semiprimes. - _Peter Munn_, Oct 28 2017
%C A202319 By arguments similar to the above proof that m = a(n)+2 is divisible by 36, it can be shown that (a(n)+2)/36 == {-1, 0, 1} (mod 5) == {-1, 0, 1} (mod 7) and that a(n) == {214, 502, 538, 718, 754, 1042, 1258} (mod 1260). - _Jon E. Schoenfield_, Feb 26 2022
%H A202319 Donovan Johnson, <a href="/A202319/b202319.txt">Table of n, a(n) for n = 1..1000</a>
%e A202319 The first twin semiprime-triple is {{213,214,215},{217,218,219}} whereby a(1)=214.
%e A202319 The fifteenth semiprime-triple is {{1891761,1891762,1891763},{1891765,1891766,1891767}} whereby a(15)=1891762.
%e A202319 The separating numbers 216 and 1891764 are divisible by 36.
%t A202319 (* Run first to define sp *) nn = 10^8; p = Prime[Range[PrimePi[nn/2]]]; lim = Floor[Sqrt[nn]]; sp = {}; k = 0; While[k++; p[[k]] <= lim, sp = Join[sp, p[[k]]*Take[p, {k, PrimePi[nn/p[[k]]]}]]]; sp = Sort[sp]; lsp = Length[sp]; b = {}; Do[If[sp[[n]] == sp[[n - 1]] + 1 && sp[[n]] == sp[[n + 1]] - 1, AppendTo[b, sp[[n - 1]]]], {n, 2, lsp-1}]; lb = Length[b]; triples = {}; Do[If[b[[i]] == b[[i - 1]] + 4, AppendTo[triples, b[[i - 1]]]], {i, 2, lb}]; triples+1 (* V.J.Pohjola, Dec 18 2011 *)
%t A202319 SequencePosition[PrimeOmega[Range[5200000]],{2,2,2,_,2,2,2}][[All,1]]+1 (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Jul 06 2017 *)
%Y A202319 Cf. A086005, A056809, A001359 Analogs: a(n) and A001359, a(n)+2 and A014574, a(n)+4 and A006512.
%K A202319 nonn
%O A202319 1,1
%A A202319 _V.J. Pohjola_, Dec 16 2011