A082875 Squares that are the sum of three factorials.
4, 9, 36, 49, 841, 5184
Offset: 1
Examples
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. The complete list of solutions is a1 a2 a3 z^2 0 0 2 4 0 1 2 4 0 2 3 9 0 4 4 49 0 5 6 841 1 1 2 4 1 2 3 9 1 4 4 49 1 5 6 841 3 3 4 36 4 5 7 5184
Programs
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Mathematica
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}] -
PARI
sum3factsq(n) = { for(a1=1,n, for(a2=a1,n, for(a3=a2,n, z = a1!+a2!+a3!; if(issquare(z),print1(z" ")) ) ) ) }
Formula
a1! + a2! + a3! = z^2.
Extensions
Sequence data ordered by Michel Marcus, Jun 03 2013