%I #15 Jan 12 2023 18:03:40

%S 0,3,15,20,44,608,22736

%N Numbers n such that triangular number t(n) (see A000217) = n(n+1)/2 is a product of three consecutive integers.

%C Replacing "three" by "two" we get A001652.

%C Replacing "three" by "N" we get: {15} for N = 4 and 5, {2079} for N = 6 and no solutions for N >= 7. - J. B. M. Melissen.

%C t(a(n)) equals x*(x+1)*(x+2) for x = [0, 1, 4, 5, 9, 56, 636]_n. - _Zak Seidov_, Jun 21 2013

%H S. P. Mohanty, <a href="http://dx.doi.org/10.1007/BF01903544">Which triangular numbers are products of three consecutive integers?</a>, Acta Mathematica Hungarica 1991, Volume 58, Issue 1-2, pp 31-36.

%t (Sqrt[8#+1]-1)/2&/@Select[Table[n(n+1)(n+2),{n,0,23000}],OddQ[Sqrt[8#+1]]&] (* _Harvey P. Dale_, Jan 12 2023 *)

%Y Cf. A000217, A001219, A001652.

%K nonn,fini,full

%O 1,2

%A _N. J. A. Sloane_, Aug 29 2004