Triples (x,y,z) with 1 < x < y < z such that x! y! z! is a square.

3, 5, 6, 4, 5, 6, 2, 7, 8, 2, 7, 9, 6, 7, 10, 2, 17, 18, 6, 19, 20, 3, 23, 24, 4, 23, 24, 3, 23, 25, 4, 23, 25, 5, 29, 30, 10, 27, 28, 2, 31, 32, 7, 34, 35, 7, 34, 36, 6, 44, 45, 3, 47, 50, 2, 48, 50, 4, 47, 50, 2, 49, 50, 3, 53, 54, 4, 53, 54, 10, 62, 63

1

This is similar to A226798, which allows x = 1. See S001001, S001002, and S001003 for subsets of this sequence. This sequence has all the triples with z - y <= 3.

T. D. Noe, Plot of 1000 triples

T. D. Noe, Table of 1000 triples

Example: (3,5,6) is here because 3! 5! 6! = 518400 = (720)^2.

A. Dujella, F. Najman, N. Saradha, and T. N. Shorey, Products of three factorials, Publ. Math. Debrecen 85/1-2 (2014), pp. 123-130.

F. Luca, N. Saradha, and T. N. Shorey, Squares and factorials in products of factorials, Monatsh Math (2014) pp. 175: 385.

(Mma) nn = 200; t = {}; Do[If[i + j + k <= nn + 3 && IntegerQ[Sqrt[i! Times @@ Range[j + 1, k] ], AppendTo[t, {i, j, k}]], {i, 2, nn}, {j, i + 1, nn}, {k, j + 1, j + 4}]; t = Sort[t, #1[[1]] + #1[[2]] + #1[[3]] < #2[[1]] + #2[[2]] + #2[[3]] &]

nonn,tabl

T. D. Noe, Apr 19 2017