A113704 Triangle read by rows. The indicator function for divisibility.

1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 0

Author

Paul Barry, Nov 05 2005

Keywords

• easy
• nonn
• tabl

Comments

From Peter Luschny, Jul 01 2023: (Start)

Definition: d divides n <=> n = m*d for some m.

Equivalently, d divides n iff d = n or d > 0, and the integer remainder of n divided by d is 0.

This definition is sufficient to define the infinite lower triangular array, i.e., if we consider only the range 0 <= d <= n. But see the construction of the inverse square array in A363914, which has to make this restriction explicit because with the above definition every integer divides 0, and thus the first row of the square matrix becomes 1 for all d.

(End)

Examples

			Triangle begins
  1;
  0, 1;
  0, 1, 1;
  0, 1, 0, 1;
  0, 1, 1, 0, 1;
  0, 1, 0, 0, 0, 1;
  0, 1, 1, 1, 0, 0, 1;
  0, 1, 0, 0, 0, 0, 0, 1;
		

References

• Tom M. Apostol, Introduction to Analytic Number Theory, Springer 1976, p. 14.

Links

• Wikipedia, Indicator function

Crossrefs

Cf. A051731, A113705 (reversed rows concatenated).

Cf. A000005 (row sums), A000007, A000961, A007947, A057427, A126988, A363914 (inverse triangle).

Programs

Maple

divides := (k, n) -> ifelse(k = n or (k > 0 and irem(n, k) = 0), 1, 0):
A113704_row := n -> local k; seq(divides(k, n), k = 0..n):
seq(print(A113704_row(n)), n = 0..9);  # Peter Luschny, Jun 28 2023

Mathematica

Table[If[k==0,Boole[n==0],Boole[Divisible[n,k]]],{n,0,10},{k,0,n}] (* Gus Wiseman, Mar 06 2020 *)

SageMath

def A113704_row(n): return [int(k.divides(n)) for k in (0..n)]
for n in (0..9): print(A113704_row(n))  # Peter Luschny, Jun 28 2023

SageMath

dim = 10
matrix(ZZ, dim, dim, lambda n, d: d <= n and ZZ(d).divides(ZZ(n)))  # Peter Luschny, Jul 01 2023

Formula

Column k has g.f. 1/(1-x^k), k >= 1. Column 0 has g.f. 1.

T(n, d) = 1 if d > 0 and d|n, otherwise 0^n. - Gus Wiseman, Mar 06 2020

Extensions

Name edited by Peter Luschny, Jul 29 2023