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A344085 Triangle of squarefree numbers first grouped by greatest prime factor, then sorted by omega, then in increasing order, read by rows.

%I A344085 #12 Feb 26 2025 20:43:57
%S A344085 1,2,3,6,5,10,15,30,7,14,21,35,42,70,105,210,11,22,33,55,77,66,110,
%T A344085 154,165,231,385,330,462,770,1155,2310,13,26,39,65,91,143,78,130,182,
%U A344085 195,273,286,429,455,715,1001,390,546,858,910,1365,1430,2002,2145,3003,5005,2730,4290,6006,10010,15015,30030
%N A344085 Triangle of squarefree numbers first grouped by greatest prime factor, then sorted by omega, then in increasing order, read by rows.
%C A344085 Differs from A339195 in having 77 before 66.
%e A344085 Triangle begins:
%e A344085    1
%e A344085    2
%e A344085    3   6
%e A344085    5  10  15  30
%e A344085    7  14  21  35  42  70 105 210
%t A344085 nn=4;
%t A344085 GatherBy[SortBy[Select[Range[Times@@Prime/@Range[nn]],SquareFreeQ[#]&&PrimePi[FactorInteger[#][[-1,1]]]<=nn&],PrimeOmega],FactorInteger[#][[-1,1]]&]
%Y A344085 Row lengths are A000079.
%Y A344085 Grouping by greatest prime factor only gives A339195.
%Y A344085 Row sums are 1 and A339360.
%Y A344085 Cf. A001221, A005117, A005183, A014466, A019565, A187769, A209862.
%Y A344085 Partition/composition orderings: A026791, A026792, A026793, A036036, A036037, A048793, A066099, A080577, A112798, A118457, A124734, A162247, A193073, A211992, A228100, A228531, A246688, A272020, A299755, A296774, A304038, A319247, A329631, A334301, A334302, A334439, A334442, A335122, A344086, A344087, A344088, A344089.
%Y A344085 Partition/composition applications: A001793, A036043, A049085, A070939, A115623, A124736, A129129, A185974, A238966, A294648, A333483, A333484, A333485, A333486, A334433, A334434, A334435, A334436, A334437, A334438, A334440, A334441, A335123, A335124.
%K A344085 nonn,tabf
%O A344085 1,2
%A A344085 _Gus Wiseman_, May 11 2021