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A216623 Triangle read by rows, n>=1, 1<=k<=n, T(n,k) = Sum_{c|n,d|k} phi(lcm(c,d)).

%I A216623 #18 Jul 28 2022 03:39:45
%S A216623 1,2,4,3,6,7,4,8,12,14,5,10,15,20,13,6,12,14,24,30,28,7,14,21,28,35,
%T A216623 42,19,8,16,24,26,40,48,56,42,9,18,19,36,45,38,63,72,37,10,20,30,40,
%U A216623 26,60,70,80,90,52,11,22,33,44,55,66,77,88,99,110,31,12,24
%N A216623 Triangle read by rows, n>=1, 1<=k<=n, T(n,k) = Sum_{c|n,d|k} phi(lcm(c,d)).
%C A216623 This is the lower triangular array of A216622, which is the main entry for this sequence.
%C A216623 T(n,1) = A000027(n).
%C A216623 T(n,n) = A062380(n).
%H A216623 Alois P. Heinz, <a href="/A216623/b216623.txt">Rows n = 1..141, flattened</a>
%e A216623 The first rows of the triangle are:
%e A216623 1,
%e A216623 2,  4,
%e A216623 3,  6,  7,
%e A216623 4,  8, 12, 14,
%e A216623 5, 10, 15, 20, 13,
%e A216623 6, 12, 14, 24, 30, 28,
%e A216623 7, 14, 21, 28, 35, 42, 19,
%e A216623 8, 16, 24, 26, 40, 48, 56, 42,
%e A216623 9, 18, 19, 36, 45, 38, 63, 72, 37,
%p A216623 with(numtheory):
%p A216623 T:= (n, k)-> add(add(phi(ilcm(c, d)), c=divisors(n)), d=divisors(k)):
%p A216623 seq (seq (T(n, k), k=1..n), n=1..14);  # _Alois P. Heinz_, Sep 12 2012
%t A216623 t[n_, k_] := Sum[ EulerPhi[ LCM[c, d]], {c, Divisors[n]}, {d, Divisors[k]}]; Table[t[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Jul 23 2013 *)
%o A216623 (Sage) # uses[A216622]
%o A216623 for n in (1..9): [A216622(n,k) for k in (1..n)]
%Y A216623 Cf. A216620, A216621, A216622, A216624, A216625, A216626, A216627.
%K A216623 nonn,tabl
%O A216623 1,2
%A A216623 _Peter Luschny_, Sep 12 2012