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A082920 Squares that are the sum of four factorials. 1
4, 9, 16, 169, 361, 729, 961, 1444, 10201, 403225, 725904 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

LINKS

Table of n, a(n) for n=0..10.

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.

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: A263094 A226354 A089149 * A204434 A110979 A073173

Adjacent sequences:  A082917 A082918 A082919 * A082921 A082922 A082923

KEYWORD

easy,nonn

AUTHOR

Cino Hilliard, May 25 2003

EXTENSIONS

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

STATUS

approved

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Last modified December 5 09:15 EST 2016. Contains 278762 sequences.