A226798 List of triples {a,b,c} with 0 < a < b < c such that a! b! c! is a square, ordered by their sum a + b + c.

1, 3, 4, 3, 5, 6, 4, 5, 6, 2, 7, 8, 2, 7, 9, 1, 8, 9, 6, 7, 10, 1, 15, 16, 2, 17, 18, 6, 19, 20, 3, 23, 24, 1, 24, 25, 4, 23, 24, 3, 23, 25, 4, 23, 25, 5, 29, 30, 10, 27, 28, 2, 31, 32, 1, 35, 36, 7, 34, 35, 7, 34, 36, 6, 44, 45, 1, 48, 49, 3, 47, 50, 2, 48, 50

Offset

1,2

Comments

Note that many of these triples have the form 1!, (k^2-1)!, (k^2)! whose product is trivially a square equal to ((k^2-1)! k)^2. [Corrected by Jonathan Sondow, Apr 01 2017]

This is a three-column table read by rows. - N. J. A. Sloane, Apr 01 2017

Links

Table of n, a(n) for n=1..75.

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

Example

The first triple is (1,3,4) because 1! 3! 4! = 144 = 12^2.

Mathematica

nn = 100; t = {}; Do[If[i + j + k <= nn + 3 && IntegerQ[Sqrt[i! j! k!]], AppendTo[t, {i, j, k}]], {i, nn}, {j, i + 1, nn}, {k, j + 1, nn}]; Sort[t, #1[[1]] + #1[[2]] + #1[[3]] < #2[[1]] + #2[[2]] + #2[[3]] &]

Crossrefs

Cf. A226799 (sums a + b + c).

Sequence in context: A205692 A342927 A363013 * A045997 A360059 A262150

Adjacent sequences: A226795 A226796 A226797 * A226799 A226800 A226801

Keyword

nonn,tabf

Author

T. D. Noe, Jun 27 2013

Status

approved