%I #7 Aug 12 2015 21:17:06

%S 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,

%T 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,

%U 12,9,8,6,0,6,0,0,0,0,0,4,13

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

%C Defined by the matrix product A054522 * A051731.

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

%e First few rows of the triangle are;

%e .1;

%e .2, 1;

%e .3, 0, 2;

%e .4, 3, 0, 2;

%e .5, 0, 0, 0, 4;

%e .6, 3, 4, 0, 0, 2;

%e .7, 0, 0, 0, 0, 0, 6;

%e .8, 7, 0, 6, 0, 0, 0, 4;

%e ....

%p A127472 := proc(n,k)

%p a := 0 ;

%p for j from k to n do

%p if (n mod j = 0 ) and (j mod k =0 ) then

%p a := a+numtheory[phi](j) ;

%p end if;

%p end do;

%p a ;

%p end proc:

%p seq(seq(A127472(n,k),k=1..n),n=1..14) ; # _R. J. Mathar_, Nov 11 2011

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

%K nonn,tabl

%O 1,2

%A _Gary W. Adamson_, Jan 15 2007