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A216624 Square array read by antidiagonals, T(n,k) = sum_{c|n,d|k} gcd(c,d) for n>=1, k>=1.

%I A216624 #23 Mar 07 2020 14:58:35
%S A216624 1,2,2,2,5,2,3,4,4,3,2,8,6,8,2,4,4,6,6,4,4,2,10,4,15,4,10,2,4,4,12,6,
%T A216624 6,12,4,4,3,11,4,16,8,16,4,11,3,4,6,8,6,8,8,6,8,6,4,2,10,10,22,4,30,4,
%U A216624 22,10,10,2,6,4,8,9,8,8,8,8,9,8,4,6
%N A216624 Square array read by antidiagonals, T(n,k) = sum_{c|n,d|k} gcd(c,d) for n>=1, k>=1.
%C A216624 T(n,k) = number of subgroups of C_n X C_k. [Hampjes et al.] - _N. J. A. Sloane_, Feb 02 2013
%H A216624 Alois P. Heinz, <a href="/A216624/b216624.txt">Antidiagonals n = 1..141, flattened</a>
%H A216624 M. Hampejs, N. Holighaus, L. Toth and C. Wiesmeyr, <a href="http://www.univie.ac.at/nuhag-php/bibtex/open_files/13485_121031-Toth-finalsubmission.pdf">On the subgroups of the group Z_m X Z_n</a>, 2012. - From _N. J. A. Sloane_, Feb 02 2013
%F A216624 T(n,n) = A060724(n) = sum_{d|n} d*tau((n/d)^2).
%F A216624 T(n,1) = T(1,n) = A000005(n) = tau(n).
%F A216624 T(n,2) = T(2,n) = A060710(n) = sum_{d|n} (3-[d is odd]) (Iverson bracket).
%F A216624 T(n+1,n) = A092517(n) = tau(n+1)*tau(n).
%F A216624 T(prime(n),1) = A007395(n) = 2.
%F A216624 T(prime(n),prime(n)) = A113935(n) = prime(n)+3.
%e A216624 [----1---2---3---4---5---6---7---8---9--10--11--12]
%e A216624 [ 1] 1,  2,  2,  3,  2,  4,  2,  4,  3,  4,  2,  6
%e A216624 [ 2] 2,  5,  4,  8,  4, 10,  4, 11,  6, 10,  4, 16
%e A216624 [ 3] 2,  4,  6,  6,  4, 12,  4,  8, 10,  8,  4, 18
%e A216624 [ 4] 3,  8,  6, 15,  6, 16,  6, 22,  9, 16,  6, 30
%e A216624 [ 5] 2,  4,  4,  6,  8,  8,  4,  8,  6, 16,  4, 12
%e A216624 [ 6] 4, 10, 12, 16,  8, 30,  8, 22, 20, 20,  8, 48
%e A216624 [ 7] 2,  4,  4,  6,  4,  8, 10,  8,  6,  8,  4, 12
%e A216624 [ 8] 4, 11,  8, 22,  8, 22,  8, 37, 12, 22,  8, 44
%e A216624 [ 9] 3,  6, 10,  9,  6, 20,  6, 12, 23, 12,  6, 30
%e A216624 [10] 4, 10,  8, 16, 16, 20,  8, 22, 12, 40,  8, 32
%e A216624 [11] 2,  4,  4,  6,  4,  8,  4,  8,  6,  8, 14, 12
%e A216624 [12] 6, 16, 18, 30, 12, 48, 12, 44, 30, 32, 12, 90
%e A216624 .
%e A216624 Displayed as a triangular array:
%e A216624 1,
%e A216624 2,  2,
%e A216624 2,  5,  2,
%e A216624 3,  4,  4,  3,
%e A216624 2,  8,  6,  8, 2,
%e A216624 4,  4,  6,  6, 4,  4,
%e A216624 2, 10,  4, 15, 4, 10, 2,
%e A216624 4,  4, 12,  6, 6, 12, 4,  4,
%e A216624 3, 11,  4, 16, 8, 16, 4, 11, 3,
%p A216624 with(numtheory):
%p A216624 T:= (n, k)-> add(add(igcd(c,d), c=divisors(n)), d=divisors(k)):
%p A216624 seq(seq(T(n, 1+d-n), n=1..d), d=1..14); # _Alois P. Heinz_, Sep 12 2012
%p A216624 T:=proc(m,n) local d; add( d*tau(m*n/d^2), d in divisors(gcd(m,n))); end; # _N. J. A. Sloane_, Feb 02 2013
%t A216624 t[n_, k_] := Sum[Sum[GCD[c, d], {c, Divisors[n]}], {d, Divisors[k]}]; Table[t[n-k+1, k], {n, 1, 12}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, May 21 2013 *)
%o A216624 (Sage)
%o A216624 def A216624(n, k) :
%o A216624     cp = cartesian_product([divisors(n), divisors(k)])
%o A216624     return reduce(lambda x,y: x+y, map(gcd, cp))
%o A216624 for n in (1..12): [A216624(n,k) for k in (1..12)]
%Y A216624 Cf. A216620, A216621, A216622, A216623, A216625, A216626, A216627.
%Y A216624 Main diagonal is A060724.
%Y A216624 Rows give A000005, A060710, A054584, A221951, A221852.
%K A216624 nonn,tabl
%O A216624 1,2
%A A216624 _Peter Luschny_, Sep 12 2012