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A294081
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Number of partitions of n into three squares and two nonnegative 7th powers.
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0
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1, 2, 3, 3, 3, 3, 3, 2, 2, 3, 4, 4, 3, 3, 3, 2, 2, 3, 5, 5, 4, 3, 3, 2, 2, 3, 5, 6, 4, 4, 3, 3, 2, 3, 5, 5, 5, 4, 5, 3, 3, 4, 5, 5, 3, 4, 4, 3, 2, 3, 6, 7, 6, 5, 6, 5, 4, 3, 4, 5, 3, 4, 4, 4, 3, 4, 7, 7, 6, 5, 5, 3, 3, 4, 7, 7, 6, 5, 4, 3, 2, 5, 7, 8, 5, 5, 6
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OFFSET
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0,2
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COMMENTS
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4^i(8j + 7) - 1^7 - 1^7 == 5 (mod 8) (when i = 0), or 2 (when i = 1), or 6 (when i >= 2). Thus, each nonnegative integer can be written as a sum of three squares and two nonnegative 7th powers; i.e., a(n) > 0.
More generally, each nonnegative integer can be written as a sum of three squares and a nonnegative k-th power and a nonnegative m-th power.
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LINKS
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EXAMPLE
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7 = 0^2 + 1^2 + 2^2 + 1^7 + 1^7 = 1^2 + 1^1 + 2^2 + 0^7 + 1^7, a(7) = 2.
10 = 0^2 + 0^2 + 3^2 + 0^7 + 1^7 = 0^2 + 1^1 + 3^2 + 0^7 + 0^7 = 0^2 + 2^2 + 2^2 + 1^7 + 1^7 = 1^2 + 2^1 + 2^2 + 0^7 + 1^7, a(10) = 4.
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MATHEMATICA
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a[n_]:=Sum[If[x^2+y^2+z^2+u^7+v^7==n, 1, 0], {x, 0, n^(1/2)}, {y, x, (n-x^2)^(1/2)}, {z, y, (n-x^2-y^2)^(1/2)}, {u, 0, (n-x^2-y^2-z^2)^(1/7)}, {v, u, (n-x^2-y^2-z^2-u^7)^(1/7)}]
Table[a[n], {n, 0, 86}]
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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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