%I #14 Oct 18 2019 00:01:56
%S 1,2,5,2,4,6,3,8,6,15,2,4,4,6,8,4,10,12,16,8,30,2,4,4,6,4,8,10,4,11,8,
%T 22,8,22,8,37,3,6,10,9,6,20,6,12,23,4,10,8,16,16,20,8,22,12,40,2,4,4,
%U 6,4,8,4,8,6,8,14,6,16,18,30,12,48,12,44,30,32
%N Triangle read by rows, n >= 1, 1 <= k <= n, T(n,k) = Sum_{c|n,d|k} gcd(c,d).
%C This is the lower triangular array of A216624, which is the main entry for this sequence.
%C T(n,1) = A000005(n) = tau(n).
%C T(n,n) = A060724(n) = Sum_{d|n} d*tau((n/d)^2).
%H Alois P. Heinz, <a href="/A216625/b216625.txt">Rows n = 1..141, flattened</a>
%e The first rows of the triangle are:
%e 1;
%e 2, 5;
%e 2, 4, 6;
%e 3, 8, 6, 15;
%e 2, 4, 4, 6, 8;
%e 4, 10, 12, 16, 8, 30;
%e 2, 4, 4, 6, 4, 8, 10;
%e 4, 11, 8, 22, 8, 22, 8, 37;
%e 3, 6, 10, 9, 6, 20, 6, 12, 23;
%p with(numtheory):
%p T:= (n, k)-> add(add(igcd(c, d), c=divisors(n)), d=divisors(k)):
%p seq (seq (T(n, k), k=1..n), n=1..14); # _Alois P. Heinz_, Sep 12 2012
%t T[n_, k_] := Sum[GCD[c, d], {c, Divisors[n]}, {d, Divisors[k]}]; Table[T[n, k], {n, 1, 14}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Mar 25 2014 *)
%o (Sage)
%o for n in (1..9): [A216624(n,k) for k in (1..n)]
%Y Cf. A216620, A216621, A216622, A216623, A216624, A216626, A216627.
%K nonn,tabl
%O 1,2
%A _Peter Luschny_, Sep 12 2012