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A094942
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Numbers having a unique partition into three squares.
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19
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0, 1, 2, 3, 4, 5, 6, 8, 10, 11, 12, 13, 14, 16, 19, 20, 21, 22, 24, 30, 32, 35, 37, 40, 42, 43, 44, 46, 48, 52, 56, 58, 64, 67, 70, 76, 78, 80, 84, 88, 91, 93, 96, 115, 120, 128, 133, 140, 142, 148, 160, 163, 168, 172, 176, 184, 190, 192, 208, 224, 232, 235
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OFFSET
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1,3
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COMMENTS
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Note that squares are allowed to be zero.
These are the numbers for which A000164(a(n)) = 1.
a(n) is the n-th largest number which has a representation as a sum of three squares (square 0 allowed), in exactly one way, if neither the order of terms nor the signs of the numbers to be squared are taken into account. The multiplicity with order and signs taken into account are A005875(a(n)).
These numbers are a proper subset of A000378.
(End)
Note that all these numbers are 4^k * A094739(n) for some k >= 0. - T. D. Noe, Nov 08 2013
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LINKS
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FORMULA
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The sequence gives the increasingly ordered members of the set {m integer | A000164(m) = 1, m >= 0}.
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EXAMPLE
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a(1) = 0 because 0 = 0^2 + 0^2 + 0^2 and 0 is the first number m with A000164(m)=1.
a(8) = 8 = 0^2 + 2^2 + 2^2, the 8th largest number m for which A000164(m) is 1.
(End)
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MATHEMATICA
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lim=25; nLst=Table[0, {lim^2}]; Do[n=a^2+b^2+c^2; If[n>0 && n<lim^2, nLst[[n]]++ ], {a, 0, lim}, {b, a, Sqrt[lim^2-a^2]}, {c, b, Sqrt[lim^2-a^2-b^2]}]; Flatten[Position[nLst, 1]]
Select[Range[0, 235], Length@PowersRepresentations[#, 3, 2] == 1 &] (* Ray Chandler, Oct 31 2019 *)
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CROSSREFS
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Cf. A025321 (numbers having a unique partition into three positive squares), A094739 (primitive n having a unique partition into three squares).
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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