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%I #12 Jun 19 2013 19:35:35
%S 6,91,946,9870,99681,996166,9992685,99991011,999872121,9999878910,
%T 99999957291,999999911791,9999993493045,99999969965911,
%U 999999863600046,9999999754307335,99999999552805215,999999998765257141,9999999993293677081,99999999982591731253,999999999933106061926
%N Largest squarefree triangular number < 10^n.
%C Number of prime factors of a(n): 2, 2, 3, 5, 3, 4, 5, 4, 5, 7, 5, 6, 4, 4, 7, 5, 6, 4, 5, 5, 5, 10, 6, 11, 4, 7, 8, 9, 8, 7. Are a(1) = 6 and a(2) = 91 the only semiprimes?
%e 6 = 2*3, 91 = 7*13.
%t t[n_] := n(n+1)/2; Table[m = 2*10^k // Sqrt // Floor; Select[Table[t[x], {x, m + 1, m - 20, -1}], SquareFreeQ[#] && # < 10^k &, 1][[1]], {k, 30}]
%t lsftr[i_]:=Module[{g=Floor[(Sqrt[1+8*10^i]-1)/2]},While[ !SquareFreeQ[ (g(g+1))/2],g--];(g(g+1))/2]; Join[{6},Array[lsftr,20,2]] (* _Harvey P. Dale_, Jun 19 2013 *)
%Y Cf. A005117 (squarefree numbers), A000217 (triangular numbers).
%Y Cf. A061304 (squarefree triangular numbers).
%K nonn
%O 1,1
%A _Zak Seidov_, Jun 05 2013
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