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A130106 A051731 * diagonalized matrix of A063659.

%I A130106 #18 Feb 20 2022 14:37:44
%S A130106 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 A130106 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 A130106 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 A130106 A051731 * diagonalized matrix of A063659.
%C A130106 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 A130106 * (1, 2, 3, ...) = A034676: (1, 5, 10, 17, 26, 50, 50, ...).
%C A130106 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 A130106 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 A130106 First few rows of the triangle:
%e A130106   1;
%e A130106   1, 2;
%e A130106   1, 0, 3;
%e A130106   1, 2, 0, 3;
%e A130106   1, 0, 0, 0, 5;
%e A130106   1, 2, 3, 0, 0, 6;
%e A130106   1, 0, 0, 0, 0, 0, 7,
%e A130106   1, 2, 0, 3, 0, 0, 0, 6;
%e A130106   1, 0, 3, 0, 0, 0, 0, 0, 8;
%e A130106   ...
%t A130106 m = 14;
%t A130106 A051731 = Table[If[Mod[n, k] == 0, 1, 0], {n, m}, {k, m}];
%t A130106 A063659 = Table[Sum[MoebiusMu[GCD[n, k]]^2, {k, n}], {n, m}] // DiagonalMatrix;
%t A130106 M = A051731.A063659;
%t A130106 Table[M[[n, k]], {n, m}, {k, n}] // Flatten (* _Jean-François Alcover_, Jan 18 2020 *)
%Y A130106 Cf. A063659, A001615 (row sums), A051731, A000010.
%K A130106 nonn,tabl
%O A130106 1,3
%A A130106 _Gary W. Adamson_, May 07 2007
%E A130106 More terms from _Jean-François Alcover_, Jan 18 2020