A130212 T(k, n) = sum_(1 <= j <= k) [j | k] j mu(k / j) floor(n / k), triangle read by rows.
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, 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, 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
Offset: 1
Examples
First few rows of the triangle are: 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, 6, 4; 9, 4, 6, 4, 4, 2, 6, 4, 6; 10, 5, 6, 4, 8, 2, 6, 4, 6, 4; ...
Programs
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Maple
with(numtheory): A130212 := (n, k) -> add(j*mobius(k / j)*iquo(n, k), j = divisors(k)); # Peter Luschny, Oct 28 2010
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Mathematica
A[n_, k_] := Sum[j MoebiusMu[k/j] Floor[n/k], {j, Divisors[k]}]; Table[A[n, k], {n, 1, 13}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 14 2019 *)
Formula
Extensions
Name replaced by new formula by Peter Luschny, Oct 28 2010
T(6,1) corrected by R. J. Mathar, Aug 06 2016