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A216625 Triangle read by rows, n >= 1, 1 <= k <= n, T(n,k) = Sum_{c|n,d|k} gcd(c,d).

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