A082920 Squares that are the sum of four factorials.

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

Offset

1,1

Links

Table of n, a(n) for n=1..11.

Example

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.

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 *)

Prog

(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)) ) ) ) ) }

Crossrefs

Cf. A082875.

Sequence in context: A226354 A299921 A089149 * A204434 A110979 A073173

Adjacent sequences: A082917 A082918 A082919 * A082921 A082922 A082923

Keyword

nonn

Author

Cino Hilliard, May 25 2003

Extensions

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

Offset corrected by Robert Israel, Dec 23 2024

Status

approved