A384447 Array read by ascending antidiagonals: A(n, k) = gcd(n, k) if n > 0 otherwise 0.

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

Offset: 0

Author

Peter Luschny, Jun 02 2025

Keywords

• nonn
• tabl

Comments

The set D = {A(n, k) : k >= 0} is a subset of [n] := {0, 1, 2,..., n} with the characteristic that for all d in D there exists a d' in D such that d*d'= n. Therefore, D may legitimately be called the 'set of divisors of n'.

However, this is not the standard definition from number theory textbooks, where an existential quantifier conjures up an infinite set out of nothing in the case n = 0. This view is also suggested by the characterization of the divisors of n as the fixed points of gcd on [n].

The form given here is constructive because it can be based on the Euclidean algorithm and with it the set of divisors is always finite.

Examples

			The array begins:
  [0] [0, 0, 0, 0, 0, 0, 0, 0, ...
  [1] [1, 1, 1, 1, 1, 1, 1, 1, ...
  [2] [2, 1, 2, 1, 2, 1, 2, 1, ...
  [3] [3, 1, 1, 3, 1, 1, 3, 1, ...
  [4] [4, 1, 2, 1, 4, 1, 2, 1, ...
  [5] [5, 1, 1, 1, 1, 5, 1, 1, ...
  [6] [6, 1, 2, 3, 2, 1, 6, 1, ...
  [7] [7, 1, 1, 1, 1, 1, 1, 7, ...
  [8] [8, 1, 2, 1, 4, 1, 2, 1, ...
  [9] [9, 1, 1, 3, 1, 1, 3, 1, ...
  ...
		

Crossrefs

Cf. A109004, A027750.

Programs

Mathematica

A[n_,k_]:=(1-KroneckerDelta[n,0])GCD[n,k]; Table[A[n-k,k],{n,0,12},{k,0,n}]//Flatten (* Stefano Spezia, Jun 02 2025 *)

Python

from math import gcd
def A(n, k): return gcd(n, k) if n > 0 else 0
for n in range(10): print([A(n, k) for k in range(8)])

Formula

A(n, k) = A109004(n, k) for 0 <= k <= n.