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A130212 T(k, n) = sum_(1 <= j <= k) [j | k] j mu(k / j) floor(n / k), triangle read by rows.

%I A130212 #14 Jun 14 2019 21:49:07
%S A130212 1,2,1,3,1,2,4,2,2,2,5,2,2,2,4,6,3,4,2,4,2,7,3,4,2,4,2,6,8,4,4,4,4,2,
%T A130212 6,4,9,4,6,4,4,2,6,4,6,10,5,6,4,8,2,6,4,6,4,11,5,6,4,8,2,6,4,6,4,10,
%U A130212 12,6,8,6,8,4,6,4,6,4,10,4,13,6,8,6,8,4,6,4,6,4,10,4,12
%N A130212 T(k, n) = sum_(1 <= j <= k) [j | k] j mu(k / j) floor(n / k), triangle read by rows.
%F A130212 A000012 * A054522 as infinite lower triangular matrices (previous name).
%F A130212 T(n,n) = A000010(n).
%e A130212 First few rows of the triangle are:
%e A130212 1;
%e A130212 2, 1;
%e A130212 3, 1, 2;
%e A130212 4, 2, 2, 2;
%e A130212 5, 2, 2, 2, 4;
%e A130212 6, 3, 4, 2, 4, 2;
%e A130212 7, 3, 4, 2, 4, 2, 6;
%e A130212 8, 4, 4, 4, 4, 2, 6, 4;
%e A130212 9, 4, 6, 4, 4, 2, 6, 4, 6;
%e A130212 10, 5, 6, 4, 8, 2, 6, 4, 6, 4;
%e A130212 ...
%p A130212 with(numtheory): A130212 := (n, k) -> add(j*mobius(k / j)*iquo(n, k), j = divisors(k)); # _Peter Luschny_, Oct 28 2010
%t A130212 A[n_, k_] := Sum[j MoebiusMu[k/j] Floor[n/k], {j, Divisors[k]}];
%t A130212 Table[A[n, k], {n, 1, 13}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Jun 14 2019 *)
%Y A130212 Cf. A000010, A130211 (product with swapped matrices), A054522, A000217 (row sums).
%K A130212 nonn,tabl,easy
%O A130212 1,2
%A A130212 _Gary W. Adamson_, May 17 2007
%E A130212 Name replaced by new formula by _Peter Luschny_, Oct 28 2010
%E A130212 T(6,1) corrected by _R. J. Mathar_, Aug 06 2016