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Number of ways to write n as x^2 + y^2 + z^2 + w^2, where w is a positive integer and x,y,z are nonnegative integers such that x + 3*y + 9*z = 2^k*m^3 for some k,m = 0,1,2,....
4

%I #14 Apr 23 2018 13:37:31

%S 1,1,1,1,2,1,1,1,2,3,1,1,2,1,2,1,4,4,1,2,1,3,1,1,4,2,4,1,2,3,1,1,4,2,

%T 4,3,2,5,4,3,3,7,3,2,3,1,3,1,3,6,7,2,4,7,3,1,5,2,6,3,2,7,7,1,4,9,3,4,

%U 2,5,5,4,5,4,6,1,5,5,1,2

%N Number of ways to write n as x^2 + y^2 + z^2 + w^2, where w is a positive integer and x,y,z are nonnegative integers such that x + 3*y + 9*z = 2^k*m^3 for some k,m = 0,1,2,....

%C Conjecture 1: a(n) > 0 for all n > 0.

%C Conjecture 2: Any positive integer can be written as x^2 + y^2 + z^2 + w^2, where x is a positive integer and y,z,w are nonnegative integers such that 2*x + 7*y = 2^k*m^3 for some k = 0,1,2 and m = 1,2,3,....

%C We have verified a(n) > 0 for all n = 1..10^7.

%C See also A301303 and A301304 for similar conjectures.

%H Zhi-Wei Sun, <a href="/A301314/b301314.txt">Table of n, a(n) for n = 1..10000</a>

%H Zhi-Wei Sun, <a href="http://dx.doi.org/10.1016/j.jnt.2016.11.008">Refining Lagrange's four-square theorem</a>, J. Number Theory 175(2017), 167-190.

%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1701.05868">Restricted sums of four squares</a>, arXiv:1701.05868 [math.NT], 2017-2018.

%e a(7) = 1 since 7 = 1^2 + 2^2 + 1^2 + 1^2 with 1 + 3*2 + 9*1 = 2*2^3.

%e a(19) = 1 since 19 = 4^2 + 1^2 + 1^2 + 1^2 with 4 + 3*1 + 9*1 = 2*2^3.

%e a(46) = 1 since 46 = 0^2 + 6^2 + 1^2 + 3^2 with 0 + 3*6 + 9*1 = 3^3.

%e a(79) = 1 since 79 = 2^2 + 7^2 + 1^2 + 5^2 with 2 + 3*7 + 9*1 = 2^2*2^3.

%e a(125) = 1 since 125 = 2^2 + 0^2 + 0^2 + 11^2 with 2 + 3*0 + 9*0 = 2*1^3.

%e a(736) = 1 since 736 = 0^2 + 24^2 + 4^2 + 12^2 with 0 + 3*24 + 9*4 = 2^2*3^3.

%t SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];

%t CQ[n_]:=CQ[n]=IntegerQ[n^(1/3)];

%t QQ[n_]:=CQ[n]||CQ[n/2]||CQ[n/4];

%t tab={};Do[r=0;Do[If[QQ[x+3y+9z]&&SQ[n-x^2-y^2-z^2],r=r+1],{x,0,Sqrt[n-1]},{y,0,Sqrt[n-1-x^2]},{z,0,Sqrt[n-1-x^2-y^2]}];tab=Append[tab,r],{n,1,80}];Print[tab]

%Y Cf. A000118, A000290, A271510, A271513, A271518, A281976, A300666, A300667, A300708, A300712, A300751, A300752, A300791, A300792, A300844, A300908, A301303, A301304.

%K nonn

%O 1,5

%A _Zhi-Wei Sun_, Mar 18 2018