A077478 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).

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, 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, 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

Offset: 1

Author

Clark Kimberling, Nov 08 2002

Keywords

• nonn
• tabl

Comments

Antidiagonal sums of R, alias row sums of T, are essentially A065608. Diagonal elements of R comprise A000203 (sums of divisors of n).

Antidiagonals of an array formed by A051731 * A051731 (transposed). - Gary W. Adamson, Nov 12 2007

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

Examples

			First few rows of the array R are:
  1, 1, 1, 1, 1, 1, 1, ...
  1, 2, 1, 2, 1, 2, 1, ...
  1, 1, 2, 1, 1, 2, 1, ...
  1, 2, 1, 3, 1, 2, 1, ...
  1, 1, 1, 1, 2, 1, 1, ...
  1, 2, 2, 2, 1, 4, 1, ...
  ...
First few rows of the triangle T are:
  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, 3, 1;
  1, 1, 2, 1, 1, 2, 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;
  ...
R(4,2)=2 since 1|2, 1|4 and 2|2, 2|4.
		

Links

• Stefano Spezia, First 150 antidiagonals of the array, flattened

Crossrefs

Cf. A051194, A077049, A077051.

Cf. A051731, A065608.

Programs

Mathematica

T[n_,k_]:=DivisorSigma[0,GCD[n,k]]; Flatten[Table[T[n-k+1,k],{n,14},{k,n}]] (* Stefano Spezia, May 23 2021 *)

Formula

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]

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

Extensions

Edited by N. J. A. Sloane, Jan 11 2009