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A229609 Array: each row starts with the least prime not in a previous row, and each prime p in a row is followed by the greatest prime < 3*p.

%I A229609 #13 Aug 17 2017 22:36:27
%S A229609 2,5,3,13,7,11,37,19,31,17,109,53,89,47,23,317,157,263,139,67,29,947,
%T A229609 467,787,409,199,83,41,2837,1399,2357,1223,593,241,113,43,8501,4177,
%U A229609 7069,3659,1777,719,337,127,59,25471,12527,21193,10973,5323,2153,1009
%N A229609 Array: each row starts with the least prime not in a previous row, and each prime p in a row is followed by the greatest prime < 3*p.
%C A229609 Conjectures: (row 1) = A126031, (column 1) = A164952, and for each row r(k), the limit of r(k)/3^k exists. For rows 1 to 4, the respective limits are 0.431270..., 0.636059..., 3.229697..., 5.015914... .
%e A229609 Northwest corner:
%e A229609    2,  5,  13,  37,  109,  317, ...
%e A229609    3,  7,  19,  53,  157,  467, ...
%e A229609   11, 31,  89, 263,  787, 2357, ...
%e A229609   17, 47, 139, 409, 1223, 3659, ...
%e A229609   23, 67, 199, 593, 1777, 5323, ...
%e A229609   29, 83, 241, 719, 2153, 6451, ...
%t A229609 seqL = 14; arr1[1] = {2}; Do[AppendTo[arr1[1], NextPrime[3*Last[arr1[1]], -1]], {seqL}];  Do[tmp = Union[Flatten[Map[arr1, Range[z]]]]; arr1[z] = {Prime[NestWhile[# + 1 &, 1, PrimePi[tmp[[#]]] - # == 0 &]]}; Do[AppendTo[arr1[z], NextPrime[3*Last[arr1[z]], -1]], {seqL}], {z, 2, 22}]; m = Map[arr1, Range[22]]; m // TableForm
%t A229609 t = Table[m[[n - k + 1]][[k]], {n, 12}, {k, n, 1, -1}] // Flatten (* _Peter J. C. Moses_, Sep 26 2013 *)
%Y A229609 Cf. A126031, A164952, A229607, A229608, A229610.
%K A229609 nonn,tabl
%O A229609 1,1
%A A229609 _Clark Kimberling_, Sep 26 2013
%E A229609 Incorrect comment deleted by _Peter Munn_, Aug 15 2017