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%I #23 Mar 08 2020 00:06:18
%S 1,2,2,2,4,2,3,4,4,3,2,6,5,6,2,4,4,6,6,4,4,2,8,4,10,4,8,2,4,4,10,6,6,
%T 10,4,4,3,8,4,12,7,12,4,8,3,4,6,8,6,8,8,6,8,6,4,2,8,8,14,4,20,4,14,8,
%U 8,2,6,4,8,9,8,8,8,8,9,8,4,6,2,12,4,12,6
%N Square array read by antidiagonals: T(n,k) = Sum_{c|n,d|k} phi(gcd(c,d)) for n>=1, k>=1.
%C T(n,n) = A060648(n) = Sum_{d|n} Dedekind_Psi(d).
%C T(n,1) = T(1,n) = A000005(n) = tau(n).
%C T(n,2) = T(2,n) = A062011(n) = 2*tau(n).
%C T(n+1,n) = A092517(n) = tau(n+1)*tau(n).
%C T(prime(n),1) = A007395(n) = 2.
%C T(prime(n),prime(n)) = A052147(n) = prime(n)+2.
%H Alois P. Heinz, <a href="/A216620/b216620.txt">Antidiagonals n = 1..141, flattened</a>
%e [----1---2---3---4---5---6---7---8---9--10--11--12]
%e [ 1] 1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 6
%e [ 2] 2, 4, 4, 6, 4, 8, 4, 8, 6, 8, 4, 12
%e [ 3] 2, 4, 5, 6, 4, 10, 4, 8, 8, 8, 4, 15
%e [ 4] 3, 6, 6, 10, 6, 12, 6, 14, 9, 12, 6, 20
%e [ 5] 2, 4, 4, 6, 7, 8, 4, 8, 6, 14, 4, 12
%e [ 6] 4, 8, 10, 12, 8, 20, 8, 16, 16, 16, 8, 30
%e [ 7] 2, 4, 4, 6, 4, 8, 9, 8, 6, 8, 4, 12
%e [ 8] 4, 8, 8, 14, 8, 16, 8, 22, 12, 16, 8, 28
%e [ 9] 3, 6, 8, 9, 6, 16, 6, 12, 17, 12, 6, 24
%e [10] 4, 8, 8, 12, 14, 16, 8, 16, 12, 28, 8, 24
%e [11] 2, 4, 4, 6, 4, 8, 4, 8, 6, 8, 13, 12
%e [12] 6, 12, 15, 20, 12, 30, 12, 28, 24, 24, 12, 50
%e .
%e Displayed as a triangular array:
%e 1,
%e 2, 2,
%e 2, 4, 2,
%e 3, 4, 4, 3,
%e 2, 6, 5, 6, 2,
%e 4, 4, 6, 6, 4, 4,
%e 2, 8, 4, 10, 4, 8, 2,
%e 4, 4, 10, 6, 6, 10, 4, 4,
%e 3, 8, 4, 12, 7, 12, 4, 8, 3,
%p with(numtheory):
%p T:= (n, k)-> add(add(phi(igcd(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_] := Outer[ EulerPhi[ GCD[#1, #2]]&, Divisors[n], Divisors[k]] // Flatten // Total; Table[ t[n-k+1, k], {n, 1, 13}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Jun 26 2013 *)
%o (Sage)
%o def A216620(n, k) :
%o cp = cartesian_product([divisors(n), divisors(k)])
%o return reduce(lambda x,y: x+y, map(euler_phi, map(gcd, cp)))
%o for n in (1..12): [A216620(n,k) for k in (1..12)]
%Y Cf. A216621, A216622, A216623, A216624, A216625, A216626, A216627.
%K nonn,tabl
%O 1,2
%A _Peter Luschny_, Sep 12 2012
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