A077478 - OEIS

%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