A051194 - OEIS

%I #32 May 20 2021 12:47:53

%S 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,

%T 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,

%U 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

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

%C 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.

%H Math StackExchange, <a href="https://math.stackexchange.com/questions/3213216/a-question-on-discrete-fourier-transform-of-some-function">A question on the discrete Fourier Transform of some function</a>

%F T(n,k) = A000005(A050873(n,k)). - _Reinhard Zumkeller_, Jun 28 2010

%F 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

%e Triangle begins:

%e {1};

%e {1,2};

%e {1,1,2};

%e {1,2,1,3};

%e {1,1,1,1,2};

%e {1,2,2,2,1,4};

%e ...

%t T[ n_, k_] := Length[Intersection[Divisors @ If[n == 0, 1, n], Divisors @ If[k == 0, 1, k]]] (* _Michael Somos_, Jul 18 2011 *)

%o (PARI) {T(n, k) = sum( i=1, min( abs(n), abs(k)),(n%i == 0) && (k%i == 0))} /* _Michael Somos_, Jul 18 2011 */

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

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

%K nonn,tabl

%O 1,3

%A _Clark Kimberling_