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%I #25 Feb 17 2021 03:58:56
%S 1,2,3,4,5,6,10,11,12,13,14,19,20,21,22,28,29,30,33,34,37,38,41,42,43,
%T 46,51,52,57,58,59,60,61,65,66,67,68,69,70,73,76,77,78,82,83,84,85,86,
%U 91,92,93,94,101,102,105,106,109,110,113,114,115,118,122,123
%N Numbers k such that the k-th triangular number is squarefree.
%C The asymptotic density of this sequence is (3/2)*A065474 = 0.4839511484... (Granville and Ramaré, 1996). - _Amiram Eldar_, Feb 17 2021
%D Steven R. Finch, Mathematical Constants II, Cambridge University Press, 2018, p. 184.
%H Amiram Eldar, <a href="/A215726/b215726.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Zak Seidov)
%H Andrew Granville and Olivier Ramaré, <a href="https://doi.org/10.1112/S0025579300011608">Explicit bounds on exponential sums and the scarcity of squarefree binomial coefficients</a>, Mathematika, Vol. 43, No. 1 (1996), pp. 73-107; <a href="https://dms.umontreal.ca/~andrew/PDF/ramare.pdf">alternative link</a>.
%F Numbers k such that A000217(k) is squarefree. [corrected by _Zak Seidov_, Jun 05 2013]
%e 14 is a term because A000217(14) = 14*15/2 = 105 = 3*5*7.
%t Select[Range[123],SquareFreeQ[#(#+1)/2]&]
%t Position[Accumulate[Range[150]],_?(SquareFreeQ[#]&)]//Flatten//Rest (* _Harvey P. Dale_, Jul 07 2020 *)
%o (PARI) is(n)=issquarefree(n/gcd(n,2))&&issquarefree((n+1)/gcd(n+1,2)) \\ _Charles R Greathouse IV_, Jun 06 2013
%Y A007674 is a subsequence.
%Y Cf. A000217, A005117, A061304, A065474.
%K nonn
%O 1,2
%A _Zak Seidov_, Aug 22 2012
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