A390838 Irregular table, where row n holds the lower left triangle of the reduced multiplication table modulo n (without row/column 0), read by rows.

1, 1, 2, 1, 1, 2, 0, 3, 2, 1, 1, 2, 4, 3, 1, 4, 4, 3, 2, 1, 1, 2, 4, 3, 0, 3, 4, 2, 0, 4, 5, 4, 3, 2, 1, 1, 2, 4, 3, 6, 2, 4, 1, 5, 2, 5, 3, 1, 6, 4, 6, 5, 4, 3, 2, 1, 1, 2, 4, 3, 6, 1, 4, 0, 4, 0, 5, 2, 7, 4, 1, 6, 4, 2, 0, 6, 4, 7, 6, 5, 4, 3, 2, 1, 1, 2, 4, 3, 6, 0, 4, 8, 3, 7, 5, 1, 6, 2, 7, 6, 3, 0, 6, 3, 0, 7, 5, 3, 1, 8, 6, 4, 8, 7, 6, 5, 4, 3, 2, 1

Offset

2,3

Comments

The rows converge to the lower left part of the usual multiplication table A003991 (without zeros), (1; 2, 4; 3, 6, 9; 4, 8, 12, 16; ...).

Length of row n is n(n-1)/2.

Row n has a zero iff n is a composite number.

Row n starts with a(n(n-1)(n-2)/6+2) = 1 and ends with (n-1, n-2, ..., 1).

Links

Table of n, a(n) for n=2..121.

Example

For n = 2, the multiplication table (mod 2) is
   x | 0 | 1 |
  ---+---+---+
   0 | 0 | 0 |
   1 | 0 | 1 |
Since the first row and column is always zero, we keep only the 1 X 1 table [1], which is the fist element of the sequence.
For n = 3, the reduced multiplication table (mod 3), without row / column 0, is:
   x | 1 | 2 |
  ---+---+---+
   1 | 1 | 2 |
   2 | 2 | 1 |   since 2 * 2 = 4 == 1 (mod 3)
The lower left triangle of the values is [ 1 ; 2, 1 ], which are the next terms (row 2) of the sequence.
For n = 4, we have
   x | 1 | 2 | 3 |
  ---+---+---+---|
   1 | 1 | 2 | 3 |
   2 | 2 | 0 | 2 |  since 2 * 2 == 0 (mod 4), 2 * 3 == 2 (mod 4).
   3 | 3 | 2 | 1 |  since 3 * 3 == 1 (mod 4).
Therefore, row 4 is [ 1 ; 2, 0 ; 3, 2, 1 ].
For n = 5, we have
   x | 1 | 2 | 3 | 4 |
  ---+---+---+---+---|
   1 | 1 |   ...     |
   2 | 2 | 4 | ...   |
   3 | 3 | 1 | 4 |...|  since 3*2 == 1 (mod 5), 3*3 == 4 (mod 5).
   4 | 4 | 3 | 2 | 1 |  since 3 * 3 == 1 (mod 4).
Therefore, row 4 is [ 1 ; 2, 4 ; 3, 1, 4 ; 4, 3, 2, 1 ].

Prog

(PARI) A390838_row(n)=concat([r*[1..r] | r <- [1..n-1]])%n

Crossrefs

Cf. A003991 (multiplication table without zeros), A004247 (multiplication table), A059692 (carryless multiplication table in base 10).

Sequence in context: A374191 A088226 A376542 * A344309 A358338 A244658

Adjacent sequences: A390835 A390836 A390837 * A390839 A390840 A390841

Keyword

nonn,base

Author

M. F. Hasler, Jan 17 2026

Status

approved