%I #19 Mar 08 2020 00:06:30
%S 1,2,2,3,4,3,4,6,6,4,5,8,7,8,5,6,10,12,12,10,6,7,12,15,14,15,12,7,8,
%T 14,14,20,20,14,14,8,9,16,21,24,13,24,21,16,9,10,18,24,28,30,30,28,24,
%U 18,10,11,20,19,26,35,28,35,26,19,20,11,12,22,30,36,40
%N Square array read by antidiagonals: T(n,k) = Sum_{c|n, d|k} phi(lcm(c,d)) for n >= 1, k >= 1.
%C T(n,n) = A062380(n) = Sum_{d|n} phi(d)*tau(d^2).
%C T(n,1) = T(1,n) = A000027(n) = n.
%C T(n,2) = T(2,n) = A005843(n) = 2*n.
%C T(n+1,n) = A002378(n) = (n+1)*n.
%C T(prime(n),1) = A000040(n) = prime(n).
%C T(prime(n),prime(n)) = 3*prime(n)-2.
%H Alois P. Heinz, <a href="/A216622/b216622.txt">Antidiagonals n = 1..141, flattened</a>
%e [-----1---2---3---4---5---6---7---8---9---10---11---12]
%e [ 1] 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12
%e [ 2] 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24
%e [ 3] 3, 6, 7, 12, 15, 14, 21, 24, 19, 30, 33, 28
%e [ 4] 4, 8, 12, 14, 20, 24, 28, 26, 36, 40, 44, 42
%e [ 5] 5, 10, 15, 20, 13, 30, 35, 40, 45, 26, 55, 60
%e [ 6] 6, 12, 14, 24, 30, 28, 42, 48, 38, 60, 66, 56
%e [ 7] 7, 14, 21, 28, 35, 42, 19, 56, 63, 70, 77, 84
%e [ 8] 8, 16, 24, 26, 40, 48, 56, 42, 72, 80, 88, 78
%e [ 9] 9, 18, 19, 36, 45, 38, 63, 72, 37, 90, 99, 76
%e [10] 10, 20, 30, 40, 26, 60, 70, 80, 90, 52, 110, 120
%e [11] 11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 31, 132
%e [12] 12, 24, 28, 42, 60, 56, 84, 78, 76, 120, 132, 98
%e .
%e Displayed as a triangular array:
%e 1,
%e 2, 2,
%e 3, 4, 3,
%e 4, 6, 6, 4,
%e 5, 8, 7, 8, 5,
%e 6, 10, 12, 12, 10, 6,
%e 7, 12, 15, 14, 15, 12, 7,
%e 8, 14, 14, 20, 20, 14, 14, 8,
%e 9, 16, 21, 24, 13, 24, 21, 16, 9,
%p with(numtheory):
%p T:= (n, k)-> add(add(phi(ilcm(c, d)), c=divisors(n)), d=divisors(k)):
%p seq (seq (T(n, 1+d-n), n=1..d), d=1..14); # _Alois P. Heinz_, Sep 12 2012
%t t[n_, k_] := Sum[ EulerPhi[LCM[c, d]], {c, Divisors[n]}, {d, Divisors[k]}]; Table[ t[n-k+1, k], {n, 1, 12}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Jun 28 2013 *)
%o (Sage)
%o def A216622(n, k) :
%o cp = cartesian_product([divisors(n), divisors(k)])
%o return reduce(lambda x,y: x+y, map(euler_phi, map(lcm, cp)))
%o for n in (1..12): [A216622(n,k) for k in (1..12)]
%Y Cf. A216620, A216621, A216623, A216624, A216625, A216626, A216627.
%K nonn,tabl
%O 1,2
%A _Peter Luschny_, Sep 12 2012