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A188621
Smallest number k>1 such that k*(n-th triangular number) is also a triangular number.
3
3, 2, 6, 12, 3, 5, 42, 56, 14, 18, 8, 10, 33, 2, 27, 240, 60, 68, 15, 3, 13, 105, 61, 67, 138, 150, 47, 51, 24, 26, 930, 117, 21, 6, 40, 66, 315, 41, 7, 231, 35, 37, 118, 5, 83, 495, 220, 230, 564, 55, 28, 147, 663, 98, 10, 50, 92, 798, 221, 229, 885, 12, 741, 615
OFFSET
1,1
COMMENTS
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.
(This is apparently a subsequence of A060544. - R. J. Mathar, Apr 06 2011)
FORMULA
a(n) = A068084(n)/A000217(n).
EXAMPLE
a(1)=3 because A000217(1)=1, 3*1 is triangular and k*1 for 1<k<3 is not triangular.
a(2)=2 because A000217(2)=3, 2*3 is triangular and k*3 for 1<k<2 (empty condition) is not triangular.
a(3)=6 because A000217(3)=6, 6*6 is triangular and k*6 for 1<k<6 is not triangular.
a(1000)=153 because A000217(1000)=500500, 153*500500=76576500 is triangular and k*500500 for 1<k<153 is not triangular.
MATHEMATICA
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 *)
snk[n_]:=Module[{k=2}, While[!OddQ[Sqrt[8k*n+1]], k++]; k]; snk/@Accumulate[ Range[ 70]] (* Harvey P. Dale, Apr 29 2018 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Zak Seidov, Apr 06 2011
STATUS
approved