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A143230 Triangle read by rows, A130207 * A000012 * A130207.

%I A143230 #12 Sep 11 2024 00:35:36
%S A143230 1,1,1,2,2,4,2,2,4,4,4,4,8,8,16,2,2,4,4,8,4,6,6,12,12,24,12,36,4,4,8,
%T A143230 8,16,8,24,16,6,6,12,12,24,12,36,24,36,4,4,8,8,16,8,24,16,24,16,10,10,
%U A143230 20,20,40,20,60,40,60,40,100,4,4,8,8,16,8,24,16,24,16,40,16
%N A143230 Triangle read by rows, A130207 * A000012 * A130207.
%C A143230 T(n,k) is the number of pairs (a,b), where 0 <= a < n, 0 <= b < k, gcd(a,n) != 1, and gcd(b,k) != 1. - _Joerg Arndt_, Jun 26 2011
%H A143230 Nathaniel Johnston, <a href="/A143230/b143230.txt">Rows 1..100, flattened</a>
%F A143230 Triangle read by rows, A130207 * A000012 * A130207, where A130207 = A000010 * 0^(n-k), 1 <= k <= n.
%F A143230 T(n,k) = phi(n) * phi(k), where phi(n) & phi(k) = Euler's totient function.
%F A143230 T(n, 0) = A000010(n) (left border).
%F A143230 Sum_{k=1..n} T(n, k) = A143231(n) (row sums).
%e A143230 First few rows of the triangle:
%e A143230   1;
%e A143230   1,  1;
%e A143230   2,  2,  4;
%e A143230   2,  2,  4,  4;
%e A143230   4,  4,  8,  8, 16;
%e A143230   2,  2,  4,  4,  8,  4;
%e A143230   6,  6, 12, 12, 24, 12, 36;
%e A143230   4,  4,  8,  8, 16,  8, 24, 16;
%e A143230   6,  6, 12, 12, 24, 12, 36, 24, 36;
%e A143230   ...
%e A143230 T(7,5) = 24 = phi(7) * phi(5) = 6 * 4.
%p A143230 with(numtheory): T := proc(n,k) return phi(n)*phi(k): end: seq(seq(T(n,k),k=1..n),n=1..12); # _Nathaniel Johnston_, Jun 26 2011
%t A143230 A143230[n_, k_]:= EulerPhi[n]*EulerPhi[k];
%t A143230 Table[A143230[n, k], {n, 12}, {k, n}] // Flatten (* _G. C. Greubel_, Sep 10 2024 *)
%o A143230 (Magma)
%o A143230 A143230:= func< n,k | EulerPhi(n)*EulerPhi(k) >;
%o A143230 [A143230(n,k): k in [1..n], n in [1..12]]; // _G. C. Greubel_, Sep 10 2024
%o A143230 (SageMath)
%o A143230 def A143230(n,k): return euler_phi(n)*euler_phi(k)
%o A143230 flatten([[A143230(n,k) for k in range(1,n+1)] for n in range(1,13)]) # _G. C. Greubel_, Sep 10 2024
%Y A143230 Cf. A000010, A130207, A143231 (row sums).
%K A143230 nonn,easy,tabl
%O A143230 1,4
%A A143230 _Gary W. Adamson_, Jul 31 2008