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A113704
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Triangle read by rows. The indicator function for divisibility.
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13
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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
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OFFSET
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0,1
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COMMENTS
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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)
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REFERENCES
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Tom M. Apostol, Introduction to Analytic Number Theory, Springer 1976, p. 14.
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LINKS
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FORMULA
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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
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EXAMPLE
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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;
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MAPLE
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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):
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MATHEMATICA
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Table[If[k==0, Boole[n==0], Boole[Divisible[n, k]]], {n, 0, 10}, {k, 0, n}] (* Gus Wiseman, Mar 06 2020 *)
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PROG
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(SageMath)
def A113704_row(n): return [int(k.divides(n)) for k in (0..n)]
(SageMath)
dim = 10
matrix(ZZ, dim, dim, lambda n, d: d <= n and ZZ(d).divides(ZZ(n))) # Peter Luschny, Jul 01 2023
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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