A267215 Integers k such that k! is the sum of 3 squares.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 50, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 79, 80

Offset: 1

Author

Altug Alkan, Jan 12 2016

Keywords

• nonn

Comments

Motivation for this sequence is the equation n! = x^2 + y^2 + z^2 where x, y and z are integers.

The asymptotic density of this sequence is 7/8 (Deshouillers and Luca, 2010). - Amiram Eldar, Jan 11 2021

Examples

			0 is a term because 0! = 1 = 0^2 + 0^2 + 1^2.
2 is a term because 2! = 2 = 0^2 + 1^2 + 1^2.
3 is a term because 3! = 6 = 1^2 + 1^2 + 2^2.
6 is a term because 6! = 720 = 0^2 + 12^2 + 24^2.
		

Links

• Jean-Marc Deshouillers and Florian Luca, How often is n! a sum of three squares?, in: The legacy of Alladi Ramakrishnan in the mathematical sciences, Springer, New York, 2010, pp. 243-251.

Crossrefs

Cf. A000408, A004215, A084953.

Complement of A084953.

Programs

Mathematica

Select[Range[0, 18], SquaresR[3, #!] > 0 &] (* Michael De Vlieger, Jan 13 2016 *)

PARI

isA004215(n) = { my(fouri, j) ; fouri=1 ; while( n >=7*fouri, if( n % fouri ==0, j= n/fouri -7 ; if( j % 8 ==0, return(1) ) ; ) ; fouri *= 4 ; ) ; return(0) ; } { for(n=1, 400, if(isA004215(n), print1(n, ", ") ; ) ; ) ; }
for(n=0, 1e2, if(!isA004215(n!), print1(n, ", ")));