A363915 Triangle read by rows. The indicator function for prime divisors, the Euclid matrix.

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

Offset

0

Comments

Definition: d prime-divides n <=> d is prime and n = m*d for some m.

Equivalently: d prime-divides n iff d is prime, and the integer remainder of n divided by d is 0. (Here the 'and' is understood as an short-circuit operator.)

This definition is sufficient to define the infinite lower triangular array, i.e., if we consider only the range 0 <= d <= n. But when looking at the square array this restriction has to be made explicit because with the above definition every integer divides 0, and thus the first row of the square matrix becomes the indicator function of the primes A010051 (with an extra zero term at the start).

References

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

Links

Paolo Xausa, Table of n, a(n) for n = 0..11475 (rows 0..150 of the triangle, flattened)

Formula

(A010051, A008472, A087624 with an extra zero term at the start).

T(n, k) = A113704(n, k) * A010051(k).

Sum_{k=0..n} k * T(n, k) = A008472(n).

Sum_{k=0..n-1} k * T(n, k) = A087624(n).

Example

Triangle T(n, k) starts:
  [0] 0;
  [1] 0, 0;
  [2] 0, 0, 1;
  [3] 0, 0, 0, 1;
  [4] 0, 0, 1, 0, 0;
  [5] 0, 0, 0, 0, 0, 1;
  [6] 0, 0, 1, 1, 0, 0, 0;
  [7] 0, 0, 0, 0, 0, 0, 0, 1;
  [8] 0, 0, 1, 0, 0, 0, 0, 0, 0;
  [9] 0, 0, 0, 1, 0, 0, 0, 0, 0, 0;
.
T(0, 0) = 0, because: [0 is prime] and [0 | 0] = False and True  = False.
T(6, 5) = 0, because: [5 is prime] and [5 | 6] = True  and False = False.
T(6, 4) = 0, because: [4 is prime] and [4 | 6] = False and False = False.
T(6, 2) = 1, because: [2 is prime] and [2 | 6] = True  and True  = True.

Maple

divides := (k, n) -> k = n or (k > 0 and irem(n, k) = 0):
T := (n, k) -> ifelse(isprime(k) and divides(k, n), 1, 0):
for n from 0 to 9 do seq(T(n, k), k = 0..n) od;

Mathematica

A363915list[rowmax_]:=Table[Boole[PrimeQ[k]&&Divisible[n, k]], {n, 0, rowmax}, {k, 0, n}]; A363915list[20] (* Generates 21 rows *) (* Paolo Xausa, Aug 04 2023 *)

Prog

(SageMath)
matrix(ZZ, 10, 10, lambda n, k: k <= n and is_prime(k) and ZZ(k).divides(ZZ(n)))

Crossrefs

Cf. A001221 (row sums), A351619 (alternating row sums), A008472, A087624.

Cf. A113704, A010051, A363914.

Sequence in context: A280710 A354353 A353669 * A353671 A353672 A323170

Adjacent sequences: A363912 A363913 A363914 * A363916 A363917 A363918

Keyword

nonn,tabl

Author

Peter Luschny, Jul 03 2023

Status

approved