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A051731 * diagonalized matrix of A063659.
1

%I #18 Feb 20 2022 14:37:44

%S 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,

%T 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,

%U 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

%N A051731 * diagonalized matrix of A063659.

%C 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, ...).

%C A130106 * (1, 2, 3, ...) = A034676: (1, 5, 10, 17, 26, 50, 50, ...).

%C 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, ...).

%F 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.

%e First few rows of the triangle:

%e 1;

%e 1, 2;

%e 1, 0, 3;

%e 1, 2, 0, 3;

%e 1, 0, 0, 0, 5;

%e 1, 2, 3, 0, 0, 6;

%e 1, 0, 0, 0, 0, 0, 7,

%e 1, 2, 0, 3, 0, 0, 0, 6;

%e 1, 0, 3, 0, 0, 0, 0, 0, 8;

%e ...

%t m = 14;

%t A051731 = Table[If[Mod[n, k] == 0, 1, 0], {n, m}, {k, m}];

%t A063659 = Table[Sum[MoebiusMu[GCD[n, k]]^2, {k, n}], {n, m}] // DiagonalMatrix;

%t M = A051731.A063659;

%t Table[M[[n, k]], {n, m}, {k, n}] // Flatten (* _Jean-François Alcover_, Jan 18 2020 *)

%Y Cf. A063659, A001615 (row sums), A051731, A000010.

%K nonn,tabl

%O 1,3

%A _Gary W. Adamson_, May 07 2007

%E More terms from _Jean-François Alcover_, Jan 18 2020