cp's OEIS Frontend

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

%I A384447 #17 Jun 03 2025 01:13:46
%S A384447 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,
%T A384447 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,
%U A384447 1,1,1,1,1,1,1,1,1,1,0,12,1,2,3,4,1,6,1,4,3,2,1,0
%N A384447 Array read by ascending antidiagonals: A(n, k) = gcd(n, k) if n > 0 otherwise 0.
%C A384447 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'.
%C A384447 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].
%C A384447 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.
%F A384447 A(n, k) = A109004(n, k) for 0 <= k <= n.
%e A384447 The array begins:
%e A384447   [0] [0, 0, 0, 0, 0, 0, 0, 0, ...
%e A384447   [1] [1, 1, 1, 1, 1, 1, 1, 1, ...
%e A384447   [2] [2, 1, 2, 1, 2, 1, 2, 1, ...
%e A384447   [3] [3, 1, 1, 3, 1, 1, 3, 1, ...
%e A384447   [4] [4, 1, 2, 1, 4, 1, 2, 1, ...
%e A384447   [5] [5, 1, 1, 1, 1, 5, 1, 1, ...
%e A384447   [6] [6, 1, 2, 3, 2, 1, 6, 1, ...
%e A384447   [7] [7, 1, 1, 1, 1, 1, 1, 7, ...
%e A384447   [8] [8, 1, 2, 1, 4, 1, 2, 1, ...
%e A384447   [9] [9, 1, 1, 3, 1, 1, 3, 1, ...
%e A384447   ...
%t A384447 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 *)
%o A384447 (Python)
%o A384447 from math import gcd
%o A384447 def A(n, k): return gcd(n, k) if n > 0 else 0
%o A384447 for n in range(10): print([A(n, k) for k in range(8)])
%Y A384447 Cf. A109004, A027750.
%K A384447 nonn,tabl
%O A384447 0,4
%A A384447 _Peter Luschny_, Jun 02 2025