A143230 - OEIS

%I #12 Sep 11 2024 00:35:36

%S 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 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 20,20,40,20,60,40,60,40,100,4,4,8,8,16,8,24,16,24,16,40,16

%N Triangle read by rows, A130207 * A000012 * A130207.

%C 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 Nathaniel Johnston, <a href="/A143230/b143230.txt">Rows 1..100, flattened</a>

%F Triangle read by rows, A130207 * A000012 * A130207, where A130207 = A000010 * 0^(n-k), 1 <= k <= n.

%F T(n,k) = phi(n) * phi(k), where phi(n) & phi(k) = Euler's totient function.

%F T(n, 0) = A000010(n) (left border).

%F Sum_{k=1..n} T(n, k) = A143231(n) (row sums).

%e First few rows of the triangle:

%e   1;

%e   1,  1;

%e   2,  2,  4;

%e   2,  2,  4,  4;

%e   4,  4,  8,  8, 16;

%e   2,  2,  4,  4,  8,  4;

%e   6,  6, 12, 12, 24, 12, 36;

%e   4,  4,  8,  8, 16,  8, 24, 16;

%e   6,  6, 12, 12, 24, 12, 36, 24, 36;

%e   ...

%e T(7,5) = 24 = phi(7) * phi(5) = 6 * 4.

%p 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[n_, k_]:= EulerPhi[n]*EulerPhi[k];

%t Table[A143230[n, k], {n, 12}, {k, n}] // Flatten (* _G. C. Greubel_, Sep 10 2024 *)

%o (Magma)

%o A143230:= func< n,k | EulerPhi(n)*EulerPhi(k) >;

%o [A143230(n,k): k in [1..n], n in [1..12]]; // _G. C. Greubel_, Sep 10 2024

%o (SageMath)

%o def A143230(n,k): return euler_phi(n)*euler_phi(k)

%o flatten([[A143230(n,k) for k in range(1,n+1)] for n in range(1,13)]) # _G. C. Greubel_, Sep 10 2024

%Y Cf. A000010, A130207, A143231 (row sums).

%K nonn,easy,tabl

%O 1,4

%A _Gary W. Adamson_, Jul 31 2008