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

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

Author

Zak Seidov, Apr 06 2011

Keywords

• nonn

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)

Examples

			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.
		

Links

• Zak Seidov, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000217, A068084, A069752, A137281.

Programs

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 *)

Formula

a(n) = A068084(n)/A000217(n).