A283617 - OEIS

%I #16 Mar 12 2017 12:34:48

%S 1,2,3,2,3,3,3,1,2,3,4,3,3,2,3,2,2,4,6,4,6,3,3,2,2,3,6,5,3,4,3,2,3,4,

%T 5,4,7,3,4,2,3,6,5,3,3,4,2,3,2,3,7,4,6,4,5,3,2,4,4,4,4,3,4,4,3,6,8,4,

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

%N Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x,y,z,w integers and 0 <= z <= w such that x^3 + 2*y^3 is a square.

%C Conjecture: (i) Let a and b >= a be positive integers with gcd(a,b) squarefree. Then, every n = 0,1,2,... can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w integers such that a*x^3 + b*y^3 is a square, if and only if (a,b) is among the ordered pairs (1,2), (1,8), (2,16), (4,23), (4,31), (5,9), (8,9), (8,225), (9,47), (25,88), (50,54).

%C (ii) Each n = 0,1,2,... can be written as x^2 + y^2 + z^2 + w^2 with x a nonnegative integer and y,z,w integers such that a*x^3 + b*y^3 is a square, whenever (a,b) is among the ordered pairs (1,8), (2,16), (8,1), (9,8), (88,25), (225,8).

%C (iii) Any positive integer can be written as x^2 + y^2 + z^2 + w^2 with x,y,z integers and w a positive integer such that x^3 + 19*y^3 + 19*z^3 is an integer cube.

%C (iv) Every n = 0,1,2,... can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w integers such that a*x^3 + b*y^3 + c*z^3 + d*w^3 is a square, whenever (a,b,c,d) is among the ordered quadruples (1,1,3,4), (1,2,3,5), (1,2,3,7), (1,2,4,5),(1,2,5,7), (1,3,4,5), (1,3,4,7), (1,3,6,8), (1,3,8,13), ((1,4,5,9), (1,7,8,11), (1,8,9,10), (1,8,9,11), (2,3,4,7), (2,4,7,9), (2,4,7,14), (2,7,9,11), (3,4,5,7), (3,8,9,11).

%C By the linked JNT paper, each n = 0,1,2,... can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w integers such that x + 2*y is a square.

%H Zhi-Wei Sun, <a href="/A283617/b283617.txt">Table of n, a(n) for n = 0..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.

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

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

%e a(79) = 1 since 79 = 3^2 + 3^2 + 5^2 + 6^2 with 3^3 + 2*3^3 = 9^2.

%e a(88) = 1 since 88 = 4^2 + 0^2 + 6^2 + 6^2 with 4^3 + 2*0^3 = 8^2.

%e a(151) = 1 since 151 = (-1)^2 + 1^2 + 7^2 + 10^2 with (-1)^3 + 2*1^3 = 1^2.

%e a(219) = 1 since 219 = 1^2 + 0^2 + 7^2 + 13^2 with 1^3 + 2*0^3 = 1^2.

%e a(438) = 1 since 438 = (-1)^2 + 1^2 + 6^2 + 20^2 with (-1)^3 + 2*1^3 = 1^2.

%e a(471) = 1 since 471 = 3^2 + (-1)^2 + 10^2 + 19^2 with 3^3 + 2*(-1)^3 = 5^2.

%e a(599) = 1 since 599 = 7^2 + (-3)^2 + 10^2 + 21^2 with 7^3 + 2*(-3)^3 = 17^2.

%e a(751) = 1 since 751 = 3^2 + 3^2 + 2^2 + 27^2 with 3^3 + 2*3^3 = 9^2.

%e a(807) = 1 since 807 = 3^2 + (-1)^2 + 11^2 + 26^2 with 3^3 + 2*(-1)^3 = 5^2.

%e a(19743) = 1 since 19743 = (-25)^2 + 25^2 + 58^2 + 123^2 with (-25)^3 + 2*25^3 = 125^2.

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

%t Do[r=0;Do[If[SQ[(-1)^i*x^3+2(-1)^j*y^3],Do[If[SQ[n-x^2-y^2-z^2],r=r+1],{z,0,Sqrt[(n-x^2-y^2)/2]}]],{x,0,Sqrt[n]},{i,0,Min[x,1]},{y,0,Sqrt[n-x^2]},{j,0,Min[y,1]}];Print[n," ",r],{n,0,80}]

%Y Cf. A000118, A000290, A000578, A271518, A281976, A283170, A283196.

%K nonn

%O 0,2

%A _Zhi-Wei Sun_, Mar 12 2017