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A282495
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Number of ways to write n as x^4 + y^2 + z^2 + w^2 with y^2 + 228*y*z + 60*z^2 a square, where x,y,z are nonnegative integers and w is a positive integer.
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6
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1, 2, 2, 2, 3, 3, 1, 1, 3, 4, 4, 3, 3, 3, 1, 1, 5, 6, 5, 5, 4, 2, 1, 2, 6, 8, 8, 4, 4, 4, 1, 2, 6, 7, 7, 6, 4, 4, 1, 3, 9, 8, 5, 3, 6, 4, 1, 2, 5, 8, 8, 7, 6, 6, 3, 3, 8, 6, 5, 4, 4, 6, 3, 2, 9, 11, 6, 6, 6, 5, 1, 3, 8, 8, 7, 4, 5, 5, 1, 3
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OFFSET
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1,2
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COMMENTS
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Conjecture: a(n) > 0 for all n > 0.
By the linked JNT paper, any nonnegative integer can be expressed as the sum of a fourth power and three squares, and each n = 0,1,2,... can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers and z*(y-z) = 0. Whether z = 0 or y = z, the number y^2 + 228*y*z + 60*z^2 is definitely a square.
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LINKS
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EXAMPLE
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a(7) = 1 since 7 = 1^4 + 1^2 + 1^2 + 2^2 with 1^2 + 228*1*1 + 60*1^2 = 17^2.
a(8) = 1 since 8 = 0^4 + 2^2 + 0^2 + 2^2 with 2^2 + 228*2*0 + 60*0^2 = 2^2.
a(15) = 1 since 15 = 1^4 + 1^2 + 3^2 + 2^2 with 1^2 + 228*1*3 + 60*3^2 = 35^2.
a(23) = 1 since 23 = 1^4 + 3^2 + 3^2 + 2^2 with 3^2 + 228*3*3 + 60*3^2 = 51^2.
a(71) = 1 since 71 = 1^4 + 5^2 + 6^2 + 3^2 with 5^2 + 228*5*6 + 60*6^2 = 95^2.
a(159) = 1 since 159 = 3^4 + 7^2 + 2^2 + 5^2 with 7^2 + 228*7*2 + 60*2^2 = 59^2.
a(623) = 1 since 623 = 3^4 + 1^2 + 10^2 + 21^2 with 1^2 + 228*1*10 + 60*10^2 = 91^2.
a(879) = 1 since 879 = 5^4 + 5^2 + 15^2 + 2^2 with 5^2 + 228*5*15 + 60*15^2 = 175^2.
a(1423) = 1 since 1423 = 1^4 + 7^2 + 2^2 + 37^2 with 7^2 + 228*7*2 + 60*2^2 = 59^2.
a(3768) = 1 since 3768 = 0^4 + 2^2 + 20^2 + 58^2 with 2^2 + 228*2*20 + 60*20^2 = 182^2.
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MATHEMATICA
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Q[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
Do[r=0; Do[If[SQ[n-x^4-y^2-z^2]&&SQ[y^2+228*y*z+60*z^2], r=r+1], {x, 0, (n-1)^(1/4)}, {y, 0, Sqrt[n-1-x^4]}, {z, 0, Sqrt[n-1-x^4-y^2]}]; Print[n, " ", r]; Continue, {n, 1, 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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