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A130106 A051731 * diagonalized matrix of A063659. 1
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,3
COMMENTS
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, ...).
LINKS
FORMULA
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.
EXAMPLE
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;
...
MATHEMATICA
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;
Table[M[[n, k]], {n, m}, {k, n}] // Flatten (* Jean-François Alcover, Jan 18 2020 *)
CROSSREFS
Cf. A063659, A001615 (row sums), A051731, A000010.
Sequence in context: A333661 A143256 A143151 * A127093 A141543 A333432
KEYWORD
nonn,tabl
AUTHOR
Gary W. Adamson, May 07 2007
EXTENSIONS
More terms from Jean-François Alcover, Jan 18 2020
STATUS
approved

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Last modified March 28 07:33 EDT 2024. Contains 371235 sequences. (Running on oeis4.)