A384446 Triangle read by rows: T(n, k) = |gcd(n, k) - k|.

0, 1, 0, 2, 0, 0, 3, 0, 1, 0, 4, 0, 0, 2, 0, 5, 0, 1, 2, 3, 0, 6, 0, 0, 0, 2, 4, 0, 7, 0, 1, 2, 3, 4, 5, 0, 8, 0, 0, 2, 0, 4, 4, 6, 0, 9, 0, 1, 0, 3, 4, 3, 6, 7, 0, 10, 0, 0, 2, 2, 0, 4, 6, 6, 8, 0, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 12, 0, 0, 0, 0, 4, 0, 6, 4, 6, 8, 10, 0

Offset: 0

Author

Peter Luschny, May 29 2025

Keywords

• nonn
• tabl

Comments

T(n, k) can be understood as a measure of the divisibility of n by k.

Examples

			Triangle starts:
  [0] [0]
  [1] [1, 0]
  [2] [2, 0, 0]
  [3] [3, 0, 1, 0]
  [4] [4, 0, 0, 2, 0]
  [5] [5, 0, 1, 2, 3, 0]
  [6] [6, 0, 0, 0, 2, 4, 0]
  [7] [7, 0, 1, 2, 3, 4, 5, 0]
  [8] [8, 0, 0, 2, 0, 4, 4, 6, 0]
  [9] [9, 0, 1, 0, 3, 4, 3, 6, 7, 0]
		

Crossrefs

Cf. A002262, A109004, A372727.

Programs

Python

from math import gcd
print([abs(gcd(n, k) - k) for n in range(13) for k in range(n + 1)])

Formula

T(n, 0) = n.

T(n, k) = 0 if k divides n, in particular T(n, 1) = T(n, n) = 0.

T(n, k) = |A002262(n, k) - A109004(n, k)|.