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

%I A051194 #32 May 20 2021 12:47:53
%S A051194 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 A051194 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 A051194 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 A051194 Triangular array T read by rows: T(n,k) = number of positive integers that divide both n and k.
%C A051194 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 A051194 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 A051194 T(n,k) = A000005(A050873(n,k)). - _Reinhard Zumkeller_, Jun 28 2010
%F A051194 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 A051194 Triangle begins:
%e A051194 {1};
%e A051194 {1,2};
%e A051194 {1,1,2};
%e A051194 {1,2,1,3};
%e A051194 {1,1,1,1,2};
%e A051194 {1,2,2,2,1,4};
%e A051194 ...
%t A051194 T[ n_, k_] := Length[Intersection[Divisors @ If[n == 0, 1, n], Divisors @ If[k == 0, 1, k]]] (* _Michael Somos_, Jul 18 2011 *)
%o A051194 (PARI) {T(n, k) = sum( i=1, min( abs(n), abs(k)),(n%i == 0) && (k%i == 0))} /* _Michael Somos_, Jul 18 2011 */
%Y A051194 Cf. A050873 (gcd), A000005 (number of divisors), A077478 (as square array).
%Y A051194 Sum of numbers in row n matches A000203. Sum of numbers in first n rows matches A024916.
%K A051194 nonn,tabl
%O A051194 1,3
%A A051194 _Clark Kimberling_