OFFSET
0,1
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)
REFERENCES
Tom M. Apostol, Introduction to Analytic Number Theory, Springer 1976, p. 14.
LINKS
Wikipedia, Indicator function
FORMULA
Column k has g.f. 1/(1-x^k), k >= 1. Column 0 has g.f. 1.
T(n,d) = 1 if d|n, otherwise 0. - Gus Wiseman, Mar 06 2020
EXAMPLE
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;
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 *)
PROG
(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
CROSSREFS
KEYWORD
AUTHOR
Paul Barry, Nov 05 2005
EXTENSIONS
Name edited by Peter Luschny, Jul 29 2023
STATUS
approved