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A143235 Triangle read by rows: T(n,k) = tau(n)*tau(k), the product of the number of divisors.

%I A143235 #13 Sep 12 2024 11:46:21
%S A143235 1,2,4,2,4,4,3,6,6,9,2,4,4,6,4,4,8,8,12,8,16,2,4,4,6,4,8,4,4,8,8,12,8,
%T A143235 16,8,16,3,6,6,9,6,12,6,12,9,4,8,8,12,8,16,8,16,12,16,2,4,4,6,4,8,4,8,
%U A143235 6,8,4,6,12,12,18,12,24,12,24,18,24,12,36,2,4,4,6,4,8,4,8,6,8,4,12,4
%N A143235 Triangle read by rows: T(n,k) = tau(n)*tau(k), the product of the number of divisors.
%C A143235 The triangle can also be created by the triple matrix product A130209 * A000012 * A130209.
%H A143235 G. C. Greubel, <a href="/A143235/b143235.txt">Rows n = 1..50 of the triangle, flattened</a>
%F A143235 T(n,k) = A000005(n)*A000005(k), for 1 <= k <= n, n >= 1.
%F A143235 Sum_{k=1..n} T(n, k) = A143236(n) (row sums).
%e A143235 First few rows of the triangle =
%e A143235   1;
%e A143235   2, 4;
%e A143235   2, 4, 4;
%e A143235   3, 6, 6,  9;
%e A143235   2, 4, 4,  6, 4;
%e A143235   4, 8, 8, 12, 8, 16;
%e A143235   2, 4, 4,  6, 4,  8, 4;
%e A143235   4, 8, 8, 12, 8, 16, 8, 16;
%e A143235   3, 6, 6,  9, 6, 12, 6, 12, 9;
%e A143235   ...
%e A143235 T(9,6) = 12 = d(9)*d(6) = 3*4.
%t A143235 A143235[n_, k_]:= DivisorSigma[0, n]*DivisorSigma[0, k];
%t A143235 Table[A143235[n, k], {n,15}, {k,n}]//Flatten (* _G. C. Greubel_, Sep 12 2024 *)
%o A143235 (Magma)
%o A143235 A143235:= func< n,k | NumberOfDivisors(n)*NumberOfDivisors(k) >;
%o A143235 [A143235(n,k): k in [1..n], n in [1..14]]; // _G. C. Greubel_, Sep 12 2024
%o A143235 (SageMath)
%o A143235 def A143235(n,k): return sigma(n,0)*sigma(k,0)
%o A143235 flatten([[A143235(n,k) for k in range(1,n+1)] for n in range(1,15)]) # _G. C. Greubel_, Sep 12 2024
%Y A143235 Cf. A000005, A035116 (right diagonal), A143236 (row sums).
%K A143235 nonn,tabl,easy
%O A143235 1,2
%A A143235 _Gary W. Adamson_, Aug 01 2008