A393746 Triangle read by rows: T(n, k) is the number of ordered pairs (i, j) of integers with 0 <= i, j < n such that i*j = k*n, for n>=1 and k=0 to n-1.

1, 3, 0, 5, 0, 0, 7, 1, 0, 0, 9, 0, 0, 0, 0, 11, 2, 2, 0, 0, 0, 13, 0, 0, 0, 0, 0, 0, 15, 2, 1, 2, 0, 0, 0, 0, 17, 1, 2, 0, 1, 0, 0, 0, 0, 19, 2, 2, 2, 2, 0, 0, 0, 0, 0, 21, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 23, 4, 4, 3, 2, 2, 2, 0, 0, 0, 0, 0, 25, 0, 0, 0, 0, 0

Offset

1,2

Comments

Row n gives the number of entries equal to k*n in the n X n multiplication table with indices 0 <= i, j < n.

Equivalently, T(n,k) counts ordered factor pairs of the multiple k*n that lie within the n X n multiplication table.

If A(n,q,r) = #{(i,j) : 0 <= i,j < n and i*j = q*n + r}, then T(n,k) = A(n,k,0), so T(n,k) is the r=0 slice of this array.

Links

Ruud H.G. van Tol, Table of n, a(n) for n = 1..10440 (144 rows)

Frédéric D. W. Heidenthal-König, Divisor-Window Formula for the Euclidean Decomposition of the Multiplication Table, Zenodo, 2026.

Formula

T(n,0) = 2*n - 1.

T(n,k) = #{(i,j) : 0 <= i,j < n and i*j = k*n}.

For k > 0, T(n,k) = #{d | k*n : d < n and (k*n)/d < n}.

For k > 0, equivalently T(n,k) = #{d | (k*n) : k < d < n}.

Sum_{k=0..n-1} T(n,k) = A018804(n).

Example

Rows for n = 1 to 12:
  n=1: 1;
  n=2: 3, 0;
  n=3: 5, 0, 0;
  n=4: 7, 1, 0, 0;
  n=5: 9, 0, 0, 0, 0;
  n=6: 11, 2, 2, 0, 0, 0;
  n=7: 13, 0, 0, 0, 0, 0, 0;
  n=8: 15, 2, 1, 2, 0, 0, 0, 0;
  n=9: 17, 1, 2, 0, 1, 0, 0, 0, 0;
  n=10: 19, 2, 2, 2, 2, 0, 0, 0, 0, 0;
  n=11: 21, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;
  n=12: 23, 4, 4, 3, 2, 2, 2, 0, 0, 0, 0, 0.

Prog

(Python)
def row(n):
    counts = [0] * n
    for i in range(n):
        for j in range(n):
            if (i * j) % n == 0:
                counts[(i * j) // n] += 1
    return counts
# triangle:
a = []
for n in range(1, 13):
    a.extend(row(n))
print(a)
(PARI) T(n, k) = sum(i=0, n-1, sum(j=0, n-1, i*j == k*n));
row(n) = vector(n, i, T(n, i-1)); \\ Michel Marcus, Mar 06 2026

Crossrefs

Cf. A004247, A018804.

Sequence in context: A136599 A227498 A131986 * A002656 A234434 A394624

Adjacent sequences: A393743 A393744 A393745 * A393747 A393748 A393749

Keyword

nonn,tabl

Author

Frédéric D. W. Heidenthal-König, Feb 26 2026

Status

approved