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A273302
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Least nonnegative integer x such that n = x^2 + y^2 + z^2 + w^2 for some nonnegative integer y,z,w with x + 3*y + 5*z a square.
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12
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0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 1, 2, 0, 0, 0, 1, 1, 2, 2, 1, 0, 0, 0, 1, 1, 0, 0, 5, 4, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 2, 4, 0, 0, 1, 1, 4, 0, 1, 0, 0, 0, 0, 5, 0, 0, 2, 0, 0, 1, 0, 1, 1, 2, 3, 0, 0, 0, 1, 0, 0, 0, 3, 4
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OFFSET
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0,16
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COMMENTS
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Clearly, a(n) = 0 if n is a square. Part (i) of the conjecture in A271518 implies that a(n) always exists.
Compare this sequence with A273294.
For more conjectural refinements of Lagrange's four-square theorem, one may consult arXiv:1604.06723.
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LINKS
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EXAMPLE
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a(6) = 1 since 6 = 1^2 + 1^2 + 0^2 + 2^2 with 1 + 3*1 + 5*0 = 2^2.
a(7) = 1 since 7 = 1^2 + 1^2 + 1^2 + 2^2 with 1 + 3*1 + 5*1 = 3^2.
a(15) = 2 since 15 = 2^2 + 3^2 + 1^2 + 1^2 with 2 + 3*3 + 5*1 = 4^2.
a(31) = 5 since 31 = 5^2 + 2^2 + 1^2 + 1^2 with 5 + 3*2 + 5*1 = 4^2.
a(32) = 4 since 32 = 4^2 + 0^2 + 0^2 + 4^2 with 4 + 3*0 + 5*0 = 2^2.
a(2384) = 24 since 2384 = 24^2 + 12^2 + 8^2 + 40^2 with 24 + 3*12 + 5*8 = 10^2.
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MATHEMATICA
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SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
Do[Do[If[SQ[n-x^2-y^2-z^2]&&SQ[x+3y+5z], Print[n, " ", x]; Goto[aa]], {x, 0, Sqrt[n]}, {y, 0, Sqrt[n-x^2]}, {z, 0, Sqrt[n-x^2-y^2]}]; Label[aa]; Continue, {n, 0, 80}]
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CROSSREFS
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Cf. A000118, A000290, A260625, A261876, A262357, A267121, A268197, A268507, A269400, A270073, A271510, A271513, A271518, A271608, A271665, A271714, A271721, A271724, A271775, A271778, A271824, A272084, A272332, A272351, A272620, A272888, A272977, A273021, A273107, A273108, A273110, A273134, A273278, A273294.
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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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