A051194 Triangular array T read by rows: T(n,k) = number of positive integers that divide both n and k.

1, 1, 2, 1, 1, 2, 1, 2, 1, 3, 1, 1, 1, 1, 2, 1, 2, 2, 2, 1, 4, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 3, 1, 2, 1, 4, 1, 1, 2, 1, 1, 2, 1, 1, 3, 1, 2, 1, 2, 2, 2, 1, 2, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 3, 1, 4, 1, 3, 2, 2, 1, 6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2

Offset: 1

Author

Clark Kimberling

Keywords

• nonn
• tabl

Comments

The function T(n,k) is defined for all integer n, k but only the values for 1<=k<=n as a triangular array are listed here.

Examples

			Triangle begins:
{1};
{1,2};
{1,1,2};
{1,2,1,3};
{1,1,1,1,2};
{1,2,2,2,1,4};
...
		

Links

• Math StackExchange, A question on the discrete Fourier Transform of some function

Crossrefs

Cf. A050873 (gcd), A000005 (number of divisors), A077478 (as square array).

Sum of numbers in row n matches A000203. Sum of numbers in first n rows matches A024916.

Programs

Mathematica

T[ n_, k_] := Length[Intersection[Divisors @ If[n == 0, 1, n], Divisors @ If[k == 0, 1, k]]] (* Michael Somos, Jul 18 2011 *)

PARI

{T(n, k) = sum( i=1, min( abs(n), abs(k)),(n%i == 0) && (k%i == 0))} /* Michael Somos, Jul 18 2011 */

Formula

T(n,k) = A000005(A050873(n,k)). - Reinhard Zumkeller, Jun 28 2010

T(n,k) = T(k,n) = T(-n,k) = T(n,-k) = T(n,n+k) = T(n+k,k). - Michael Somos, Jul 18 2011