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A261876
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Number of ordered ways to write n as x^2 + y^2 + z^2 + w^2 with (5*x^2+7*y^2+9*z^2)*y*z a square, where x,y,z,w are nonnegative integers with z > 0.
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23
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1, 3, 2, 1, 4, 5, 1, 3, 5, 5, 4, 2, 4, 7, 2, 1, 9, 9, 4, 4, 7, 5, 1, 5, 6, 12, 7, 1, 10, 9, 2, 3, 10, 9, 7, 5, 4, 11, 3, 5, 14, 10, 4, 4, 10, 9, 3, 2, 8, 17, 10, 4, 11, 18, 6, 7, 9, 6, 11, 2, 10, 15, 4, 1, 15, 17, 4, 9, 13, 10
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OFFSET
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1,2
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COMMENTS
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Conjecture: (i) a(n) > 0 for all n > 0, and a(n) = 1 only for n = 4^k*m (k = 0,1,2,... and m = 1, 7, 23, 647, 863).
(ii) For each triple (a,b,c) = (1,8,20), (3,5,15), (6,14,4), (7,29,5), (18,38,18), (39,81,51), (42,98,14), any natural number can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that x*y*(a*x^2+b*y^2+c*z^2) is a square.
For more refinements of Lagrange's four-square theorem, see arXiv:1604.06723.
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LINKS
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EXAMPLE
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a(4) = 1 since 4 = 0^2 + 0^2 + 2^2 + 0^2 with 2 > 0 and (5*0^2+7*0^2+9*2^2)*0*2 = 0^2.
a(7) = 1 since 7 = 2^2 + 1^2 + 1^2 + 1^2 with 1 > 0 and (5*2^2+7*1^2+9*1^2)*1*1 = 6^2.
a(23) = 1 since 23 = 2^2 + 1^2 + 3^2 + 3^2 with 3 > 0 and (5*2^2+7*1^2+9*3^2)*1*3 = 18^2.
a(647) = 1 since 647 = 13^2 + 1^2 + 6^2 + 21^2 with 6 > 0 and (5*13^2+7*1^2+9*6^2)*1*6 = 84^2.
a(863) = 1 since 863 = 1^2 + 23^2 + 18^2 + 3^2 with 18 > 0 and (5*1^2+7*23^2+9*18^2)*23*18 = 1656^2.
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MATHEMATICA
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SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
Do[r=0; Do[If[SQ[n-x^2-y^2-z^2]&&SQ[y*z(5x^2+7y^2+9z^2)], r=r+1], {x, 0, Sqrt[n-1]}, {y, 0, Sqrt[n-1-x^2]}, {z, 1, Sqrt[n-x^2-y^2]}]; Print[n, " ", r]; Continue, {n, 1, 70}]
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CROSSREFS
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Cf. A000118, A000290, A260625, A262357, A267121, A268507, A269400, A271510, A271513, A271518, A271608, A271665, A271714, A271721, A271724, A271775, A271778, A271824, A272084, A272332, A272351.
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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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