A127481 Triangle T(n,k) read by rows: T(n,k) = sum_{l=k..n, l|n, k|l} l*phi(k).

1, 3, 2, 4, 0, 6, 7, 6, 0, 8, 6, 0, 0, 0, 20, 12, 8, 18, 0, 0, 12, 8, 0, 0, 0, 0, 0, 42, 15, 14, 0, 24, 0, 0, 0, 32, 13, 0, 24, 0, 0, 0, 0, 0, 54, 18, 12, 0, 0, 60, 0, 0, 0, 0, 40, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 110, 28, 24, 42, 32, 0, 36, 0, 0, 0, 0, 0, 48, 14, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Author

Gary W. Adamson, Jan 15 2007

Keywords

• nonn
• tabl
• easy

Examples

			First few rows of the triangle are:
1;
3, 2;
4, 0, 6;
7, 6, 0, 8;
6, 0, 0, 0, 20,
12, 8, 18, 0, 0, 12;
8, 0, 0, 0, 0, 0, 42;
15, 14, 0, 24, 0, 0, 0, 32;
...
		

Crossrefs

Cf. A054522, A127093, A001157 (row sums), A002618, A127466.

Programs

Maple

A127481 := proc(n,k)
    a :=0 ;
    for l from k to n do
        if modp(n,l) =0 and modp(l,k) =0 then
            a := a+l*numtheory[phi](k) ;
        end if;
    end do:
    a ;
end proc: # R. J. Mathar, Sep 06 2013

Formula

T(n,1) = A000203(n).

T(n,n) = A002618(n).

T(n,k) =sum_{l=k..n} A127093(n,l) * A054522(l,k), the matrix product of the infinite lower triangular matrices.