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A095678 Minimal sequence such that all triples of consecutive numbers have no common divisor greater than 1, but all three pairs within the triples are not coprime.

%I A095678 #19 Sep 29 2024 02:53:01
%S A095678 6,10,15,12,20,45,18,40,75,24,50,135,36,80,225,48,100,375,54,160,405,
%T A095678 72,200,675,96,250,1125,108,320,1215,144,400,1875,162,500,2025,192,
%U A095678 640,3375,216,800,3645,288,1000,5625,324,1250,6075,384,1280,9375,432
%N A095678 Minimal sequence such that all triples of consecutive numbers have no common divisor greater than 1, but all three pairs within the triples are not coprime.
%H A095678 Amiram Eldar, <a href="/A095678/b095678.txt">Table of n, a(n) for n = 1..10000</a>
%F A095678 gcd(a(n),a(n+1),a(n+2)) = 1, gcd(a(n),a(n+1)) > 1, gcd(a(n),a(n+2)) > 1 and gcd(a(n+1),a(n+2)) > 1.
%F A095678 A001221(a(n)) = 2; 2 <= A020639(a(n)) <= 3 <= A006530(a(n)) <= 5.
%F A095678 From _Jianing Song_, Jun 08 2022: (Start)
%F A095678 a(3n-2) = A033845(n) = 6*A003586(n);
%F A095678 a(3n-1) = A033846(n) = 10*A003592(n);
%F A095678 a(3n) = A033849(n) = 15*A003593(n). (End)
%F A095678 Sum_{n>=1} 1/a(n) = 7/8. - _Amiram Eldar_, Sep 29 2024
%t A095678 seq1[p_, q_, lim_] := Sort[Flatten[Table[p^i * q^j, {i, 1, Log[p, lim]}, {j, 1, Log[q, lim/p^i]}]]];
%t A095678 seq[lim_] := Module[{s1 = seq1[2, 3, lim], s2 = seq1[2, 5, lim], s3 = seq1[3, 5, lim], ns}, ns = Length[s3]; Flatten[Transpose[{s1[[1;;ns]], s2[[1;;ns]], s3}]]]; seq[10^4] (* _Amiram Eldar_, Sep 29 2024 *)
%Y A095678 Cf. A001221, A003586, A003592, A003593, A006530, A020639, A033845, A033846, A033849.
%K A095678 nonn
%O A095678 1,1
%A A095678 _Reinhard Zumkeller_, Jul 04 2004