A082920 Squares that are the sum of four factorials.

4, 9, 16, 169, 361, 729, 961, 1444, 10201, 403225, 725904

Offset: 1

Author

Cino Hilliard, May 25 2003

Keywords

• nonn

Examples

			These appear to be the only solutions to a! + b! + c! + d! = n^2:
a b c d n
0 0 0 0 4
0 0 0 1 4
0 0 0 3 9
0 0 1 1 4
0 0 1 3 9
0 1 1 1 4
0 1 1 3 9
0 2 3 6 729
0 4 4 5 169
0 4 8 9 403225
0 5 5 5 361
0 5 5 6 961
0 5 7 7 10201
1 1 1 1 4
1 1 1 3 9
1 2 3 6 729
1 4 4 5 169
1 4 8 9 403225
1 5 5 5 361
1 5 5 6 961
1 5 7 7 10201
2 2 3 3 16
2 2 6 6 1444
4 5 9 9 725904
1!+2!+3!+6! = 729 = 27^2. This shows that 4 factorials can add to a cube.
		

Crossrefs

Cf. A082875.

Programs

Maple

S:= {}:
N:= 100: # for terms < (N+1)!
F:= [seq(i!,i=1..N)]:
for a from 1 to N do
  for b from a to N do
   for c from b to N do
     for d from c to N do
     if issqr(F[a]+F[b]+F[c]+F[d]) then
       S:= S union {F[a]+F[b]+F[c]+F[d]}
fi od od od od:
sort(convert(S,list)); # Robert Israel, Dec 23 2024

Mathematica

e = 75; a = Union[ Flatten[ Table[a! + b! + c! + d!, {a, 1, e}, {b, a, e}, {c, b, e}, {d, c, e}]]]; l = Length[a]; Do[ If[ IntegerQ[ Sqrt[ a[[i]] ]], Print[ a[[i]] ]], {i, 1, l}]
Select[Union[Total/@Tuples[Range[10]!,4]],IntegerQ[Sqrt[#]]&] (* Harvey P. Dale, Aug 23 2014 *)

PARI

sum4factsq(n) = { for(a1=0,n, for(a2=a1,n, for(a3=a2,n, for(a4=a3,n, z = a1!+a2!+a3!+a4!; if(issquare(z),print(a1" "a2" "a3" "a4" "z)) ) ) ) ) }

Extensions

Edited, corrected and extended by Robert G. Wilson v, May 26 2003

Offset corrected by Robert Israel, Dec 23 2024