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%I #39 May 24 2021 00:45:46
%S 1,1,1,1,2,1,1,1,1,1,1,2,2,2,1,1,1,1,1,1,1,1,2,1,3,1,2,1,1,1,2,1,1,2,
%T 1,1,1,2,1,2,2,2,1,2,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,1,4,1,3,2,2,1,1,1,
%U 1,1,1,1,1,1,1,1,1,1,1,2,1,2,1,2,2,2,1,2,1,2,1,1,1,2,1,2,2,1,1,2,2,1,2,1,1
%N Rectangular array R read by antidiagonals: R(i,j) is the number of integers k that divide both i and j (i >= 1, j >= 1).
%C Antidiagonal sums of R, alias row sums of T, are essentially A065608. Diagonal elements of R comprise A000203 (sums of divisors of n).
%C Antidiagonals of an array formed by A051731 * A051731 (transposed). - _Gary W. Adamson_, Nov 12 2007
%C If R(n) is the n X n Redheffer matrix (A143104) and Rt(n) is its transposed matrix, then this sequence seems to be formed by R(n)*Rt(n). - _Enrique Pérez Herrero_, Feb 21 2012
%H Stefano Spezia, <a href="/A077478/b077478.txt">First 150 antidiagonals of the array, flattened</a>
%F R=U*V, where U and V are the summatory matrices (A077049, A077051). The triangle T(n, k) formed by antidiagonals: T(n, k)=tau(gcd(k, n+1-k)) for 1<=k<=n, where tau(m)=A000005(m). [Corrected by _Leroy Quet_, Apr 08 2009]
%F Dirichlet g.f.: Sum_{n>=1} Sum_{k>=1} tau(gcd(n,k))/n^s/k^c = zeta(s)*zeta(c)* zeta(s + c). - _Mats Granvik_, May 19 2021
%e First few rows of the array R are:
%e 1, 1, 1, 1, 1, 1, 1, ...
%e 1, 2, 1, 2, 1, 2, 1, ...
%e 1, 1, 2, 1, 1, 2, 1, ...
%e 1, 2, 1, 3, 1, 2, 1, ...
%e 1, 1, 1, 1, 2, 1, 1, ...
%e 1, 2, 2, 2, 1, 4, 1, ...
%e ...
%e First few rows of the triangle T are:
%e 1;
%e 1, 1;
%e 1, 2, 1;
%e 1, 1, 1, 1;
%e 1, 2, 2, 2, 1;
%e 1, 1, 1, 1, 1, 1;
%e 1, 2, 1, 3, 1, 3, 1;
%e 1, 1, 2, 1, 1, 2, 1, 1;
%e 1, 2, 1, 2, 2, 2, 1, 2, 1;
%e 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
%e 1, 2, 2, 3, 1, 4, 1, 3, 2, 2, 1;
%e ...
%e R(4,2)=2 since 1|2, 1|4 and 2|2, 2|4.
%t T[n_,k_]:=DivisorSigma[0,GCD[n,k]]; Flatten[Table[T[n-k+1,k],{n,14},{k,n}]] (* _Stefano Spezia_, May 23 2021 *)
%Y Cf. A051194, A077049, A077051.
%Y Cf. A051731, A065608.
%K nonn,tabl
%O 1,5
%A _Clark Kimberling_, Nov 08 2002
%E Edited by _N. J. A. Sloane_, Jan 11 2009
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