A267215 - OEIS

%I #23 Jan 11 2021 02:50:47

%S 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,

%T 30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,50,51,52,53,55,

%U 56,57,58,59,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,79,80

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

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

%C The asymptotic density of this sequence is 7/8 (Deshouillers and Luca, 2010). - _Amiram Eldar_, Jan 11 2021

%H Jean-Marc Deshouillers and Florian Luca, <a href="https://doi.org/10.1007/978-1-4419-6263-8_14">How often is n! a sum of three squares?</a>, in: The legacy of Alladi Ramakrishnan in the mathematical sciences, Springer, New York, 2010, pp. 243-251.

%e 0 is a term because 0! = 1 = 0^2 + 0^2 + 1^2.

%e 2 is a term because 2! = 2 = 0^2 + 1^2 + 1^2.

%e 3 is a term because 3! = 6 = 1^2 + 1^2 + 2^2.

%e 6 is a term because 6! = 720 = 0^2 + 12^2 + 24^2.

%t Select[Range[0, 18], SquaresR[3, #!] > 0 &] (* _Michael De Vlieger_, Jan 13 2016 *)

%o (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, ", ") ; ) ; ) ; }

%o for(n=0, 1e2, if(!isA004215(n!), print1(n, ", ")));

%Y Cf. A000408, A004215, A084953.

%Y Complement of A084953.

%K nonn

%O 1,3

%A _Altug Alkan_, Jan 12 2016