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

1,11

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).

Links

Table of n, a(n) for n=1..81.

Formula

For n > 0:

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

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

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

Example

   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

Prog

(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)]

Crossrefs

Sequence in context: A352080 A295632 A139549 * A280569 A140345 A177706

Adjacent sequences: A216912 A216913 A216914 * A216916 A216917 A216918

Keyword

nonn,tabl

Author

Peter Luschny, Oct 02 2012

Status

approved