A127472 Triangle T(n,k) = Sum_{j=k..n, j|n, k|j} phi(j) read by rows, 1<=k<=n.

1, 2, 1, 3, 0, 2, 4, 3, 0, 2, 5, 0, 0, 0, 4, 6, 3, 4, 0, 0, 2, 7, 0, 0, 0, 0, 0, 6, 8, 7, 0, 6, 0, 0, 0, 4, 9, 0, 8, 0, 0, 0, 0, 0, 6, 10, 5, 0, 0, 8, 0, 0, 0, 0, 4, 11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 12, 9, 8, 6, 0, 6, 0, 0, 0, 0, 0, 4, 13

Offset: 1

Author

Gary W. Adamson, Jan 15 2007

Keywords

• nonn
• tabl

Comments

Defined by the matrix product A054522 * A051731.

Examples

			First few rows of the triangle are;
.1;
.2, 1;
.3, 0, 2;
.4, 3, 0, 2;
.5, 0, 0, 0, 4;
.6, 3, 4, 0, 0, 2;
.7, 0, 0, 0, 0, 0, 6;
.8, 7, 0, 6, 0, 0, 0, 4;
....
		

Crossrefs

Cf. A054522, A051731, A062949 (row sums), A000010 (diagonal n=k), A127471 (swapped matrix product).

Programs

Maple

A127472 := proc(n,k)
        a := 0 ;
        for j from k to n do
                if (n mod j = 0 ) and (j mod k =0 ) then
                        a := a+numtheory[phi](j) ;
                end if;
        end do;
        a ;
end proc:
seq(seq(A127472(n,k),k=1..n),n=1..14) ; # R. J. Mathar, Nov 11 2011

Formula

T(n,k) = Sum_{j=k..n} A054522(n,j) * A051731(j,k), 1<=k<=n.