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A216621 Triangle read by rows, n >= 1, 1 <= k <= n, T(n,k) = Sum_{c|n,d|k} phi(gcd(c,d)).

%I A216621 #15 Oct 18 2019 00:01:05
%S A216621 1,2,4,2,4,5,3,6,6,10,2,4,4,6,7,4,8,10,12,8,20,2,4,4,6,4,8,9,4,8,8,14,
%T A216621 8,16,8,22,3,6,8,9,6,16,6,12,17,4,8,8,12,14,16,8,16,12,28,2,4,4,6,4,8,
%U A216621 4,8,6,8,13,6,12,15,20,12,30,12,28,24,24,12
%N A216621 Triangle read by rows, n >= 1, 1 <= k <= n, T(n,k) = Sum_{c|n,d|k} phi(gcd(c,d)).
%C A216621 This is the lower triangular array of A216620, which is the main entry for this sequence.
%C A216621 T(n,1) = A000005(n) = tau(n).
%C A216621 T(n,n) = A060648(n) = sum{d|n} Dedekind_Psi(d).
%H A216621 Alois P. Heinz, <a href="/A216621/b216621.txt">Rows n = 1..141, flattened</a>
%e A216621 The first rows of the triangle are:
%e A216621   1;
%e A216621   2,  4;
%e A216621   2,  4,  5;
%e A216621   3,  6,  6, 10;
%e A216621   2,  4,  4,  6,  7;
%e A216621   4,  8, 10, 12,  8, 20;
%e A216621   2,  4,  4,  6,  4,  8,  9;
%e A216621   4,  8,  8, 14,  8, 16,  8, 22;
%e A216621   3,  6,  8,  9,  6, 16,  6, 12, 17;
%e A216621   4,  8,  8, 12, 14, 16,  8, 16, 12, 28;
%e A216621   2,  4,  4,  6,  4,  8,  4,  8,  6,  8, 13;
%p A216621 with(numtheory):
%p A216621 T:= (n, k)-> add(add(phi(igcd(c,d)), c=divisors(n)), d=divisors(k)):
%p A216621 seq (seq (T(n, k), k=1..n), n=1..14);  # _Alois P. Heinz_, Sep 12 2012
%t A216621 t[n_, k_] := Sum[ EulerPhi[GCD[c, d]], {c, Divisors[n]}, {d, Divisors[k]}]; Table[t[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Jun 28 2013 *)
%o A216621 (Sage)
%o A216621 for n in (1..9): [A216620(n,k) for k in (1..n)]
%Y A216621 Cf. A216620, A216622, A216623, A216624, A216625, A216626, A216627.
%K A216621 nonn,tabl
%O A216621 1,2
%A A216621 _Peter Luschny_, Sep 12 2012