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A188621 Smallest number k>1 such that k*(n-th triangular number) is also a triangular number.

%I A188621 #32 Apr 29 2018 21:26:08
%S A188621 3,2,6,12,3,5,42,56,14,18,8,10,33,2,27,240,60,68,15,3,13,105,61,67,
%T A188621 138,150,47,51,24,26,930,117,21,6,40,66,315,41,7,231,35,37,118,5,83,
%U A188621 495,220,230,564,55,28,147,663,98,10,50,92,798,221,229,885,12,741,615
%N A188621 Smallest number k>1 such that k*(n-th triangular number) is also a triangular number.
%C A188621 There is a sequence of triangular numbers >3 which are not divisible by any smaller triangular number > 1, primitive triangular numbers in that sense: 3, 10, 28, 55, 91, 136, 253.... whose indices are in A137281.
%C A188621 (This is apparently a subsequence of A060544. - R. J. Mathar, Apr 06 2011)
%H A188621 Zak Seidov, <a href="/A188621/b188621.txt">Table of n, a(n) for n = 1..1000</a>
%F A188621 a(n) = A068084(n)/A000217(n).
%e A188621 a(1)=3 because A000217(1)=1, 3*1 is triangular and k*1 for 1<k<3 is not triangular.
%e A188621 a(2)=2 because A000217(2)=3, 2*3 is triangular and k*3 for 1<k<2 (empty condition) is not triangular.
%e A188621 a(3)=6 because A000217(3)=6, 6*6 is triangular and k*6 for 1<k<6 is not triangular.
%e A188621 a(1000)=153 because A000217(1000)=500500, 153*500500=76576500 is triangular and k*500500 for 1<k<153 is not triangular.
%t A188621 TriangularQ[n_] := IntegerQ[Sqrt[1 + 8 n]]; Table[t = (n + 1)*n/2; k = 2; While[! TriangularQ[k*t], k++]; k, {n, 100}] (* _T. D. Noe_, Apr 06 2011 *)
%t A188621 snk[n_]:=Module[{k=2},While[!OddQ[Sqrt[8k*n+1]],k++];k]; snk/@Accumulate[ Range[ 70]] (* _Harvey P. Dale_, Apr 29 2018 *)
%Y A188621 Cf. A000217, A068084, A069752, A137281.
%K A188621 nonn
%O A188621 1,1
%A A188621 _Zak Seidov_, Apr 06 2011