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A350857
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Number of ways to write n as 2*w^4 + x^4 + y^2 + z^2, where w,x,y,z are nonnegative integers with w or x a square.
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7
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1, 2, 3, 3, 3, 3, 2, 2, 2, 2, 3, 3, 2, 2, 1, 1, 3, 3, 5, 4, 5, 3, 2, 2, 1, 3, 5, 4, 4, 3, 1, 2, 4, 4, 6, 4, 5, 5, 4, 2, 3, 6, 5, 5, 2, 3, 2, 2, 3, 3, 7, 4, 7, 6, 3, 3, 2, 3, 5, 3, 1, 4, 2, 2, 3, 5, 6, 5, 7, 4, 3, 2, 2, 4, 5, 3, 3, 3, 1, 1, 2, 6, 8, 9, 5, 7, 5, 3, 4, 3, 6, 6, 4, 3, 2, 0, 3, 6, 7, 5, 7
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OFFSET
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0,2
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COMMENTS
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Conjecture 1: a(n) > 0 except for n = 95, 255.
This has been verified for n up to 10^7.
It seems that a(n) = 1 only for n = 0, 14, 15, 24, 30, 60, 78, 79, 111, 159, 174, 249, 270, 303, 318, 334, 382, 399, 414, 462, 472, 831, 894, 975, 1080, 1151, 1164, 1919, 4080, 6456, 20319.
Conjecture 2: For any positive integer a == 2 (mod 4), all sufficiently large integers can be written as a*w^4 + x^4 + (2*y)^2 + z^2 with w,x,y,z integers. If N(a) denotes the largest integer not of the form a*w^4 + x^4 + (2*y)^2 + z^2 (w,x,y,z = 0,1,2,...), then we have N(2) = 255, N(6) = 2716, N(10) = 598, N(14) = 8427, N(18) = 2463, N(22) = 3884, N(26) = 14988, N(30) = 10843.
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LINKS
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EXAMPLE
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a(24) = 1 with 24 = 2*0^4 + 2^4 + 2^2 + 2^2 and 0 = 0^2.
a(1151) = 1 with 1151 = 2*3^4 + 4^4 + 2^2 + 27^2 and 4 = 2^2.
a(6456) = 1 with 6456 = 2*1^4 + 3^4 + 17^2 + 78^2 and 1 = 1^2.
a(20319) = 1 with 20319 = 2*5^4 + 0^4 + 5^2 + 138^2 and 0 = 0^2.
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MATHEMATICA
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SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
tab={}; Do[r=0; Do[If[SQ[n-2w^4-x^4-y^2]&&(SQ[w]||SQ[x]), r=r+1], {w, 0, (n/2)^(1/4)}, {x, 0, (n-2w^4)^(1/4)}, {y, 0, Sqrt[(n-2w^4-x^4)/2]}]; tab=Append[tab, r], {n, 0, 100}]; Print[tab]
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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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