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

Author

Gary W. Adamson, May 17 2007

Keywords

• nonn
• tabl
• easy

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;
...
		

Crossrefs

Cf. A000010, A130211 (product with swapped matrices), A054522, A000217 (row sums).

Programs

Maple

with(numtheory): A130212 := (n, k) -> add(j*mobius(k / j)*iquo(n, k), j = divisors(k)); # Peter Luschny, Oct 28 2010

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

A000012 * A054522 as infinite lower triangular matrices (previous name).

T(n,n) = A000010(n).

Extensions

Name replaced by new formula by Peter Luschny, Oct 28 2010

T(6,1) corrected by R. J. Mathar, Aug 06 2016