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Express n as the sum of four squares, x^2+y^2+z^2+w^2, with x>=y>=z>=w>=0, maximizing the value of x. Then a(n) is that x.
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%I #23 Feb 26 2024 19:17:09

%S 0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,3,4,5,5,5,5,5,5,5,4,5,

%T 5,5,6,6,6,6,6,6,6,5,6,6,6,6,6,7,7,7,7,7,7,7,6,7,7,7,7,7,7,7,8,8,8,8,

%U 8,8,8,7,8,8,8,8,8,8,8,7,8,9,9,9,9,9,9,9,8,9,9,9,9,9,9,9,8,9,9,9,10,10,10,10,10,10,10,9,10,10,10,10,10,10,10,9,10

%N Express n as the sum of four squares, x^2+y^2+z^2+w^2, with x>=y>=z>=w>=0, maximizing the value of x. Then a(n) is that x.

%C Lagrange's theorem tells us that each positive integer can be written as a sum of four squares.

%H David Consiglio, Jr., <a href="/A178786/b178786.txt">Table of n, a(n) for n = 0..10000</a>

%H David Consiglio, Jr., <a href="/A178786/a178786.txt">Python program</a>

%o (Python)

%o from math import *

%o for nbre in range(0, 500): # or more than 500 !

%o maxc4=0

%o for c1 in range(0, int(sqrt(nbre/4))+1):

%o for c2 in range(c1, int(sqrt(nbre/3))+1):

%o for c3 in range(c2, int(sqrt(nbre/2))+1):

%o s3=c3**2+c2**2+c1**2

%o if s3<=nbre:

%o c4=sqrt(nbre-s3)

%o if int(c4)==c4 and c4>=c3:

%o if c4>maxc4:

%o maxc4=int(c4)

%o print(maxc4, end=', ')

%Y Cf. A122922, A122923, A122924, A122925, A122926, A122927, A002330, A122921.

%Y Analogs for 3 squares: A261904 and A261915.

%K nonn

%O 0,5

%A _Sébastien Dumortier_, Jun 24 2011