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

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

Offset: 1

Author

Gary W. Adamson, Jul 31 2008

Keywords

• nonn
• easy
• tabl

Comments

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

Examples

			First few rows of the triangle:
  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,  8, 16,  8, 24, 16;
  6,  6, 12, 12, 24, 12, 36, 24, 36;
  ...
T(7,5) = 24 = phi(7) * phi(5) = 6 * 4.
		

Links

• Nathaniel Johnston, Rows 1..100, flattened

Crossrefs

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

Programs

Magma

A143230:= func< n,k | EulerPhi(n)*EulerPhi(k) >;
[A143230(n,k): k in [1..n], n in [1..12]]; // G. C. Greubel, Sep 10 2024

Maple

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

Mathematica

A143230[n_, k_]:= EulerPhi[n]*EulerPhi[k];
Table[A143230[n, k], {n, 12}, {k, n}] // Flatten (* G. C. Greubel, Sep 10 2024 *)

SageMath

def A143230(n,k): return euler_phi(n)*euler_phi(k)
flatten([[A143230(n,k) for k in range(1,n+1)] for n in range(1,13)]) # G. C. Greubel, Sep 10 2024

Formula

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

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

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

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