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A308453
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Number of integer-sided triangles with perimeter n whose smallest side length is squarefree.
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1
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0, 0, 1, 0, 1, 1, 2, 1, 3, 2, 4, 2, 4, 2, 5, 3, 6, 5, 8, 6, 10, 8, 12, 9, 13, 10, 14, 10, 14, 11, 15, 12, 17, 14, 19, 16, 21, 18, 24, 20, 26, 23, 29, 25, 32, 28, 35, 31, 38, 34, 42, 37, 45, 40, 48, 42, 51, 45, 54, 47, 56, 49, 59, 52, 62, 56, 66, 59, 70, 63
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OFFSET
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1,7
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LINKS
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FORMULA
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a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} sign(floor((i+k)/(n-i-k+1))) * mu(k)^2, where mu is the Möbius function (A008683).
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EXAMPLE
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There exist A005044(12) = 3 integer-sided triangles with perimeter = 12; these three triangles have respectively sides: (2, 5, 5), (3, 4, 5) or (4, 4, 4). Only the last one: (4, 4, 4) has a smallest side length = 4 that is not squarefree, so a(12) = 2. - Bernard Schott, Jan 22 2023
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MATHEMATICA
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Table[Sum[Sum[ MoebiusMu[k]^2* Sign[Floor[(i + k)/(n - i - k + 1)]], {i, k, Floor[(n - k)/2]}], {k, Floor[n/3]}], {n, 100}]
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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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