A216915 T(n, k) = Product{1<=j<=n, gcd(j,k)=1 | j} / lcm{1<=j<=n, gcd(j,k)=1 | j} for n >= 0, k >= 1, square array read by antidiagonals.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 12, 1, 2, 1, 1, 1, 1, 12, 1, 2, 1, 1, 1, 1, 1, 48, 1, 2, 1, 2, 1, 1, 1, 1, 144, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1440, 3, 8, 1, 12, 1, 2, 1, 1, 1, 1, 1440, 3, 8, 1, 12, 1, 2, 1, 1, 1, 1, 1, 17280, 3, 80

Offset: 1

Author

Peter Luschny, Oct 02 2012

Keywords

• nonn
• tabl

Comments

T(n,k) = Product(R(n,k))/lcm(R(n,k)) where R(n,k) is the set of all integers up to n that are relatively prime to k.

T(n,k) = A216919(n,k)/A216917(n,k).

Examples

			   k |n=0  1  2  3  4  5  6  7  8   9   10
  ---+------------------------------------
   1 |  1  1  1  1  2  2 12 12 48 144 1440
   2 |  1  1  1  1  1  1  1  1  1   3    3
   3 |  1  1  1  1  2  2  2  2  8   8   80
   4 |  1  1  1  1  1  1  1  1  1   3    3
   5 |  1  1  1  1  2  2 12 12 48 144  144
   6 |  1  1  1  1  1  1  1  1  1   1    1
   7 |  1  1  1  1  2  2 12 12 48 144 1440
   8 |  1  1  1  1  1  1  1  1  1   3    3
   9 |  1  1  1  1  2  2  2  2  8   8   80
  10 |  1  1  1  1  1  1  1  1  1   3    3
  11 |  1  1  1  1  2  2 12 12 48 144 1440
  12 |  1  1  1  1  1  1  1  1  1   1    1
  13 |  1  1  1  1  2  2 12 12 48 144 1440
		

Programs

Sage

def A216915(n, k):
    def R(n, k): return [j for j in (1..n) if gcd(j, k) == 1]
    return mul(R(n,k))/lcm(R(n, k))
for k in (1..13): [A216915(n, k) for n in (0..10)]

Formula

For n > 0:

A(n,1) = A025527(n);

A(4,n) = A000034(n);

A(n,n) = A128247(n).