A394016 Triangle read by rows: T(n, k) = 1 if k appears in the n X n multiplication table, otherwise 0.

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

Offset

1,1

Comments

The zeros correspond to integers that cannot be written as a product of two numbers <= n. These holes reflect gaps in divisor pairs within the n X n box.

Let A(n,q,r) = #{(i,j) : 1 <= i,j <= n and i*j = q*n + r}, with 0 <= r < n. Then T(n,k) = 1 iff A(n,floor(k/n), k mod n) > 0.

Links

Paolo Xausa, Table of n, a(n) for n = 1..14910 (rows 1..35 of the triangle, flattened).

Frédéric D. W. Heidenthal-König, The Support of the Euclidean Decomposition of the Multiplication Table, Zenodo, 2026.

Frédéric D. W. Heidenthal-König, Motion and Animation in Multiplication, Zenodo, 2026.

Frédéric D. W. Heidenthal-König, Trace across Bases, Zenodo, 2026.

Frédéric D. W. Heidenthal-König, Square-grid display of the positive-indexed support sequence for n=256.

Frédéric D. W. Heidenthal-König, Interactive visualization of the support rows arranged as n X n bitmaps.

Formula

T(n,k) = 1 if k = a*b for some integers 1 <= a,b <= n, otherwise 0, for 1 <= k <= n^2.

Equivalently, T(n,k) = 1 iff k has a divisor d with 1 <= d <= n and k/d <= n.

Sum_{k=1..n^2} T(n,k) = #{a*b : 1 <= a,b <= n} = A027424(n).

Sum_{k=1..n^2} k * T(n,k) = A321165(n). - Alois P. Heinz, Mar 06 2026

Example

Rows for n = 1 to 6:
  n=1: 1;
  n=2: 1, 1, 0, 1;
  n=3: 1, 1, 1, 1, 0, 1, 0, 0, 1;
  n=4: 1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1;
  n=5: 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1;
  n=6: 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1.

Maple

T:= (n, k)-> `if`(ormap(d-> max(d, k/d)<=n, numtheory[divisors](k)), 1, 0):
seq(seq(T(n, k), k=1..n^2), n=1..6);  # Alois P. Heinz, Mar 06 2026

Mathematica

A394016[n_, k_] := Boole[AnyTrue[Divisors[k], # <= n && k <= n*# &]];
Table[A394016[n, k], {n, 6}, {k, n^2}] (* Paolo Xausa, Mar 29 2026 *)

Prog

(Python)
def row(n):
    r = []
    for k in range(1, n*n + 1):
        found = 0
        for a in range(1, n+1):
            if k % a == 0 and k//a <= n:
                found = 1
                break
        r.append(found)
    return r
# triangle
a = []
for n in range(1, 6):
    a.extend(row(n))
print(a)
(Python)
import numpy as np
n = 256
row = np.zeros(n*n, dtype=np.uint8)
for a in range(1, n+1):
    row[a-1:a*n:a] = 1 # memory stride
grid = row.reshape(n, n)

Crossrefs

Cf. A027424, A321165.

Sequence in context: A175087 A318924 A356162 * A253414 A127872 A129564

Adjacent sequences: A394013 A394014 A394015 * A394017 A394018 A394019

Keyword

nonn,tabf,changed

Author

Frédéric D. W. Heidenthal-König, Mar 06 2026

Status

approved