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A061304 Squarefree triangular numbers.

%I A061304 #19 Apr 17 2020 07:18:36
%S A061304 1,3,6,10,15,21,55,66,78,91,105,190,210,231,253,406,435,465,561,595,
%T A061304 703,741,861,903,946,1081,1326,1378,1653,1711,1770,1830,1891,2145,
%U A061304 2211,2278,2346,2415,2485,2701,2926,3003,3081,3403,3486,3570,3655,3741,4186,4278
%N A061304 Squarefree triangular numbers.
%H A061304 Zak Seidov, <a href="/A061304/b061304.txt">Table of n, a(n) for n = 1..1000</a>
%F A061304 A010054(a(n))*A008966(a(n)) = 1. - _Reinhard Zumkeller_, Nov 01 2009
%F A061304 a(n) = A000217(A215726(n)). - _Zak Seidov_, Aug 22 2012
%e A061304 105 = 3 * 5 * 7 is a squarefree triangular number.
%p A061304 # uses code of A000217
%p A061304 isA061304 := proc(n)
%p A061304     isA000217(n) and issqrfree(n) ;
%p A061304     simplify(%) ;
%p A061304 end proc:
%p A061304 for n from 1 to 5000 do
%p A061304     if isA061304(n) then
%p A061304         printf("%d,",n);
%p A061304     end if;
%p A061304 end do: # _R. J. Mathar_, Oct 05 2017
%t A061304 Select[Accumulate[Range[0, 100]], SquareFreeQ] (* _Jean-François Alcover_, Apr 17 2020 *)
%o A061304 (PARI) isA078779F(f)=for(i=2,#f~, if(f[i,2]>1, return(0))); #f~==0 || f[1,2]==1 || (f[1,2]==2 && f[1,1]==2)
%o A061304 list(lim)=my(v=List(), ok=1); forfactored(n=2, (sqrtint(lim\1*8+1)+1)\2, e=n[2][,2]; if(isA078779F(n[2]), if(ok, listput(v, binomial(n[1],2)), ok=1), ok=0)); Vec(v) \\ _Charles R Greathouse IV_, Nov 05 2017
%Y A061304 Cf. A000217, A215726.
%K A061304 nonn,easy
%O A061304 1,2
%A A061304 _Amarnath Murthy_, Apr 26 2001