A397409 Irregular triangle read by rows: row n lists, in increasing order, the products n*k such that A033677(n*k) = n.

1, 2, 4, 3, 6, 9, 8, 12, 16, 5, 10, 15, 20, 25, 18, 24, 30, 36, 7, 14, 21, 28, 35, 42, 49, 32, 40, 48, 56, 64, 27, 45, 54, 63, 72, 81, 50, 60, 70, 80, 90, 100, 11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 121, 84, 96, 108, 120, 132, 144, 13, 26, 39, 52, 65, 78, 91, 104, 117, 130, 143, 156, 169

Offset

1,2

Comments

Row n contains the positive multiples n*k for which n is the smallest divisor of n*k such that n >= sqrt(n*k). This forces 1 <= k <= n, because n >= sqrt(n*k) implies n^2 >= n*k; i.e., k <= n; and k >= 1 by definition, since n*k is a positive multiple of n.

For n > 1 the length of row n is A062854(n) = A333995(n) + A010051(n) = A027424(n) - A027424(n-1); row 1 has length 1.

Every positive integer N occurs in exactly one row, namely the row A033677(N). Thus the rows partition the positive integers N according to the value of A033677(N). Thus, the entire sequence is a permutation of the positive integers.

The initial term of row n is D(n); i.e., the smallest primitive entry in column n of the triangular multiplication table. The last term of row n is always n^2, because A033677(n^2) = n.

This triangle is distinct from A075383 where the n-th row has always exactly n terms. Here, row n consists exactly of the products that first appear when passing from the (n-1) X (n-1) multiplication table to the n X n multiplication table. The row first terms agree with A075384 for n = 1..39, but first differ at n = 40, where the row initial term here is 880 while A075384(40) = 920.

Links

Table of n, a(n) for n=1..72.

Formula

Row n = {x: A033677(x) = n} in increasing order.

Equivalently, row n = {n*k: 1 <= k <= n, and A033677(n*k) = n} in increasing order.

The row initial term is D(n) = n*rho(n), where rho(n) = min{k >= 1: A033677(n*k) = n}.

Example

The first rows are:
  row 1: 1;
  row 2: 2,  4;
  row 3: 3,  6, 9;
  row 4:     8, 12, 16;
  row 5: 5, 10, 15, 20, 25;
  row 6:        18, 24, 30, 36;
  row 7: 7, 14, 21, 28, 35, 42, 49;
  row 8:            32, 40, 48, 56, 64;
  row 9:        27,     45, 54, 63, 72, 81;
For row 4, the product 4 = 4*1 is not listed because A033677(4) = 2, so 4 first appears in row 2. The products 8, 12, and 16 are listed because A033677(8) = A033677(12) = A033677(16) = 4. Hence 8 is the start of row 4.
For row 6, the products 6 = 6*1 and 12 = 6*2 are not listed because A033677(6) = 3 and A033677(12) = 4. The products 18, 24, 30, and 36 are listed because A033677(18) = A033677(24) = A033677(30) = A033677(36) = 6. Hence 18 is the start of row 6.

Mathematica

row[n_] := Select[n * Range[n], Function[x, Select[Divisors[x], # >= Sqrt[x] &, 1] == {n}]]; Table[row[n], {n, 1, 13}] // Flatten (* Amiram Eldar, Jun 27 2026 *)

Prog

(PARI) f(n) = my(d=divisors(n)); d[length(d)\2+1]; \\ A033677
row(n) = select(x->f(x)==n, vector(n, k, k*n)); \\ Michel Marcus, Jul 05 2026

Crossrefs

Cf. A010051, A027424, A033676, A033677, A062854, A075383, A075384, A095163, A095164, A333995.

Cf. A000290 (right border).

Sequence in context: A331018 A054428 A095167 * A075383 A338919 A348018

Adjacent sequences: A397406 A397407 A397408 * A397411 A397412 A397414

Keyword

nonn,tabf,new

Author

Charles Kusniec, Jun 23 2026

Status

approved