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A113704 Triangle read by rows. The indicator function for divisibility. 13
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 (list; table; graph; refs; listen; history; text; internal format)
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
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
Cf. A051731, A113705 (reversed rows concatenated).
Cf. A000005 (row sums), A000007, A000961, A007947, A057427, A126988, A363914 (inverse triangle).
Sequence in context: A177990 A287722 A284588 * A341150 A244220 A283963
KEYWORD
easy,nonn,tabl
AUTHOR
Paul Barry, Nov 05 2005
EXTENSIONS
Name edited by Peter Luschny, Jul 29 2023
STATUS
approved

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Last modified March 28 20:05 EDT 2024. Contains 371254 sequences. (Running on oeis4.)