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1, 1, 2, 1, 0, 3, 1, 2, 0, 3, 1, 0, 0, 0, 5, 1, 2, 3, 0, 0, 6, 1, 0, 0, 0, 0, 0, 7, 1, 2, 0, 3, 0, 0, 0, 6, 1, 0, 3, 0, 0, 0, 0, 0, 8, 1, 2, 0, 0, 5, 0, 0, 0, 0, 10, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 1, 2, 3, 3, 0, 6, 0, 0, 0, 0, 0, 9, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 13, 1, 2, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 14
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OFFSET
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1,3
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COMMENTS
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Right border = A063659, (1, 2, 3, 3, 5, 6, 7, 6, 8, 10, ...), the Moebius transform of A001615: (1, 3, 4, 6, 6, 12, 8, 12, 12, ...).
A130106 * (1, 2, 3, ...) = A034676: (1, 5, 10, 17, 26, 50, 50, ...).
A034676^(-1) * (1,2,3,...) = 1/1, 1/2, 2/3, 2/3, 4/5, 2/6, 6/7, 4/6, 6/8, 4/10, ...; where the numerators = phi(n), A000010: (1, 1, 2, 2, 4, 2, 6, 4, ...); and the denominators = A063659, the right border of the triangle: (1, 2, 3, 3, 5, 6, 7, 8, 10, ...).
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LINKS
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FORMULA
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Inverse Moebius transform of an infinite lower triangular matrix with A063659, (1, 2, 3, 3, 5, 6, 7, 6, 8, 10, ...) in the main diagonal and the rest zeros.
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EXAMPLE
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First few rows of the triangle:
1;
1, 2;
1, 0, 3;
1, 2, 0, 3;
1, 0, 0, 0, 5;
1, 2, 3, 0, 0, 6;
1, 0, 0, 0, 0, 0, 7,
1, 2, 0, 3, 0, 0, 0, 6;
1, 0, 3, 0, 0, 0, 0, 0, 8;
...
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MATHEMATICA
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m = 14;
A051731 = Table[If[Mod[n, k] == 0, 1, 0], {n, m}, {k, m}];
A063659 = Table[Sum[MoebiusMu[GCD[n, k]]^2, {k, n}], {n, m}] // DiagonalMatrix;
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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