A102715 Triangle read by rows: T(n,k) is phi(binomial(n,k)), where phi is Euler's totient function (0 <= k <= n).

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 2, 2, 1, 1, 4, 4, 4, 4, 1, 1, 2, 8, 8, 8, 2, 1, 1, 6, 12, 24, 24, 12, 6, 1, 1, 4, 12, 24, 24, 24, 12, 4, 1, 1, 6, 12, 24, 36, 36, 24, 12, 6, 1, 1, 4, 24, 32, 48, 72, 48, 32, 24, 4, 1, 1, 10, 40, 80, 80, 120, 120, 80, 80, 40, 10, 1, 1, 4, 20, 80, 240, 240

Offset: 0

Author

Emeric Deutsch, Feb 06 2005

Keywords

• nonn
• tabl

Comments

Row n contains n+1 terms. Row sums yield A064450.

Examples

			T(6,3)=8 because the positive integers relatively prime to binomial(6,3)=20 and not exceeding 20 are 1,3,7,9,11,13,17 and 19.
Triangle begins:
  1;
  1, 1;
  1, 1, 1;
  1, 2, 2, 1;
  1, 2, 2, 2, 1;
  1, 4, 4, 4, 4, 1;
		

Crossrefs

Cf. A000010 (totient), A007318 (binomial).

Cf. A064450, A066973.

Programs

Magma

/* As triangle */ [[EulerPhi(Binomial(n,k)): k in [0..n]]: n in [0.. 10]]; // Vincenzo Librandi, May 01 2019

Maple

with(numtheory): T:=(n,k)->phi(binomial(n,k)): for n from 0 to 12 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form

Mathematica

Flatten[Table[EulerPhi[Binomial[n, k]], {n, 0, 12}, {k, 0, n}]] (* Vincenzo Librandi, May 01 2019 *)

Formula

T(n, k) = A000010(A007318(n, k)) (0 <= k <= n).

T(2n,n) = A066973(n).