A152005 Numbers whose square is the product of two distinct tetrahedral numbers A000292.

2, 140, 280, 1092, 166460, 189070, 665840, 804540, 845460, 34250920, 38336088, 133784560, 138535992, 225792840, 4998790160, 6301258040, 7559616818, 8367691640, 39991371446, 104637102152, 227490888350, 1497809326860, 296523233581822

Offset

1,1

Comments

There may be values that are not given in the recurrence shown. This sequence is suggested by Ulas, p. 11, who supplied the recurrence.

a(24) > 3*10^14. - Donovan Johnson, Jan 11 2012

Links

Table of n, a(n) for n=1..23.

Maciej Ulas, On certain Diophantine equations related to triangular and tetrahedral numbers, arXiv:0811.2477 [math.NT], 2008.

Formula

a(n) = T(i)*T(j) where T(k) = A000292(k) = C(k+2,3) = k*(k+1)*(k+2)/6.

Example

From R. J. Mathar, Jan 22 2009: (Start)
2 is in the sequence because 2^2 = 4*1 = T(2)*T(1).
140 is in the sequence 140^2 = 560*35 = T(14)*T(5) = 19600*1 = T(48)*T(1).
280 is in the sequence because 280^2 = 19600*4 = T(48)*T(2).
1092 is in the sequence because 1092^2 = 3276*364 = T(26)*T(12). (End)

Mathematica

(* This program is not suitable to compute more than a dozen terms. *)
terms = 12; imin = 1; imax = 3000;
Union[Reap[Do[k2 = i(i+1)(i+2)/6 j(j+1)(j+2)/6; k = Sqrt[k2]; If[IntegerQ[k], Print[k]; Sow[k]], {i, imin, imax}, {j, i+1, imax}]][[2, 1]]][[1 ;; terms]] (* Jean-François Alcover, Oct 31 2018 *)

Crossrefs

Cf. A000292, A175497 (products distinct triangular numbers).

Sequence in context: A265176 A101232 A093887 * A140898 A220513 A120814

Adjacent sequences: A152002 A152003 A152004 * A152006 A152007 A152008

Keyword

nonn,more

Author

Jonathan Vos Post, Nov 19 2008

Extensions

Sequence replaced by sequence with no intermediate terms missing by R. J. Mathar, Jan 22 2009

a(15)-a(18) from Donovan Johnson, Jan 24 2009

a(19)-a(23) from Donovan Johnson, Jan 11 2012

Status

approved