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A327567
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Number of r X s rectangles with squarefree side lengths such that r <= s, r + s = 2n and r | s.
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2
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1, 2, 2, 2, 1, 3, 2, 2, 2, 3, 2, 3, 1, 3, 2, 2, 2, 4, 2, 4, 4, 4, 1, 3, 0, 3, 2, 2, 2, 5, 2, 1, 3, 4, 3, 4, 2, 3, 3, 3, 1, 6, 2, 4, 3, 3, 2, 3, 2, 1, 3, 4, 2, 4, 3, 3, 4, 4, 1, 6, 1, 4, 3, 1, 2, 5, 2, 3, 3, 6, 2, 4, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 5, 2, 3, 3, 3, 2, 6
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OFFSET
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1,2
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COMMENTS
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a(n) is the number of divisors r of 2*n such that r <= n and r and n-r are squarefree. - Robert Israel, Apr 27 2020
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LINKS
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FORMULA
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a(n) = Sum_{i=1..n} mu(i)^2 * mu(2*n-i)^2 * (1-ceiling((2*n-i)/i)+floor((2*n-i)/i)), where mu is the Möbius function (A008683).
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EXAMPLE
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a(6) = 3; 2*6 = 12 has three rectangles with squarefree side lengths, 1 X 11, 2 X 10 and 6 X 6, such that 1 | 11, 2 | 10 and 6 | 6.
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MAPLE
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f:= proc(n) uses numtheory;
nops(select(r -> r <= n and issqrfree(r) and issqrfree(2*n-r), divisors(2*n), n))
end proc:
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MATHEMATICA
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Table[Sum[MoebiusMu[i]^2 MoebiusMu[2 n - i]^2 (1 - Ceiling[(2 n - i)/i] + Floor[(2 n - i)/i]), {i, n}], {n, 100}]
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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