%I #9 Oct 01 2013 17:57:38
%S 4,9,36,49,841,5184
%N Squares that are the sum of three factorials.
%F a1! + a2! + a3! = z^2.
%e These appear to be the only solutions. 8 and 27 appear to be the only cubes that are the sum of 3 factorials. Again, it appears that 2 and 3 are the only powers of n satisfying a1!+a2!+a3! = z^n.
%e The complete list of solutions is
%e a1 a2 a3 z^2
%e 0 0 2 4
%e 0 1 2 4
%e 0 2 3 9
%e 0 4 4 49
%e 0 5 6 841
%e 1 1 2 4
%e 1 2 3 9
%e 1 4 4 49
%e 1 5 6 841
%e 3 3 4 36
%e 4 5 7 5184
%t d = 50; a = Union[ Flatten[ Table[a! + b! + c!, {a, 1, d}, {b, a, d}, {c, b, d}]]]; l = Length[a]; Do[ If[ IntegerQ[ Sqrt[ a[[i]]]], Print[ a[[i]]]], {i, 1, l}]
%o (PARI) sum3factsq(n) = { for(a1=1,n, for(a2=a1,n, for(a3=a2,n, z = a1!+a2!+a3!; if(issquare(z),print1(z" ")) ) ) ) }
%Y Cf A114377, A162681.
%K easy,nonn
%O 1,1
%A _Cino Hilliard_, May 25 2003
%E Sequence data ordered by _Michel Marcus_, Jun 03 2013