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A301304
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Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x >= y >= 0 <= z <= w such that x^2 + 7*y^2 = 2^k*m for some k = 0,1,2 and m = 1,2,3,....
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4
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1, 2, 2, 2, 2, 3, 1, 2, 3, 4, 3, 4, 2, 3, 2, 2, 4, 5, 3, 4, 4, 2, 1, 3, 2, 6, 5, 4, 3, 4, 2, 2, 6, 5, 4, 6, 3, 3, 3, 4, 6, 8, 2, 5, 5, 3, 2, 4, 4, 5, 6, 5, 5, 4, 3, 3, 6, 5, 2, 6, 3, 4, 3, 2, 6, 10, 3, 5, 8, 1, 2, 5, 5, 6, 5, 6, 5, 3, 1, 4
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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.
We have verified this for n up to 10^8.
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LINKS
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EXAMPLE
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a(79) = 1 since 79 = 5^2 + 1^2 + 2^2 + 7^2 with 5^2 + 7*1^2 = 2^2*2^3.
a(323) = 1 since 323 = 3^2 + 1^2 + 12^2 + 13^2 with 3^2 + 7*1^2 = 2*2^3.
a(646) = 1 since 646 = 22^2 + 11^2 + 4^2 + 5^2 with 22^2 + 7*11^2 = 11^3.
a(815) = 1 since 815 = 9^2 + 5^2 + 15^2 + 22^2 with 9^2 + 7*5^2 = 2^2*4^3.
a(1111) = 1 since 1111 = 1^2 + 1^2 + 22^2 + 25^2 with 1^2 + 7*1^2 = 2^3.
a(2822) = 1 since 2822 = 2^2 + 0^2 + 3^2 + 53^2 with 2^2 + 7*0^2 = 2^2*1^3.
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MATHEMATICA
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SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
CQ[n_]:=CQ[n]=IntegerQ[n^(1/3)];
QQ[n_]:=n>0&&(CQ[n]||CQ[n/2]||CQ[n/4]);
tab={}; Do[r=0; Do[If[QQ[x^2+7y^2], Do[If[SQ[n-x^2-y^2-z^2], r=r+1], {z, 0, Sqrt[(n-x^2-y^2)/2]}]], {y, 0, Sqrt[n/2]}, {x, y, Sqrt[n-y^2]}]; tab=Append[tab, r], {n, 1, 80}]; Print[tab]
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CROSSREFS
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Cf. A000118, A000290, A271510, A271513, A271518, A281976, A300666, A300667, A300708, A300712, A300751, A300752, A300791, A300792, A300844, A300908, A301303, A301314.
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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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