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A283617
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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.
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2
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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, 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, 9, 6, 2, 3, 4, 4, 7, 5, 4, 4, 3, 1, 3
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OFFSET
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0,2
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COMMENTS
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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).
(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).
(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.
(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).
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.
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LINKS
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EXAMPLE
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a(0) = 1 since 0 = 0^2 + 0^2 + 0^2 + 0^2 with 0^3 + 2*0^3 = 0^2.
a(7) = 1 since 7 = (-1)^2 + 1^2 + 1^2 + 2^2 with (-1)^3 + 2*1^3 = 1^2.
a(79) = 1 since 79 = 3^2 + 3^2 + 5^2 + 6^2 with 3^3 + 2*3^3 = 9^2.
a(88) = 1 since 88 = 4^2 + 0^2 + 6^2 + 6^2 with 4^3 + 2*0^3 = 8^2.
a(151) = 1 since 151 = (-1)^2 + 1^2 + 7^2 + 10^2 with (-1)^3 + 2*1^3 = 1^2.
a(219) = 1 since 219 = 1^2 + 0^2 + 7^2 + 13^2 with 1^3 + 2*0^3 = 1^2.
a(438) = 1 since 438 = (-1)^2 + 1^2 + 6^2 + 20^2 with (-1)^3 + 2*1^3 = 1^2.
a(471) = 1 since 471 = 3^2 + (-1)^2 + 10^2 + 19^2 with 3^3 + 2*(-1)^3 = 5^2.
a(599) = 1 since 599 = 7^2 + (-3)^2 + 10^2 + 21^2 with 7^3 + 2*(-3)^3 = 17^2.
a(751) = 1 since 751 = 3^2 + 3^2 + 2^2 + 27^2 with 3^3 + 2*3^3 = 9^2.
a(807) = 1 since 807 = 3^2 + (-1)^2 + 11^2 + 26^2 with 3^3 + 2*(-1)^3 = 5^2.
a(19743) = 1 since 19743 = (-25)^2 + 25^2 + 58^2 + 123^2 with (-25)^3 + 2*25^3 = 125^2.
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MATHEMATICA
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SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
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}]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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