A392445 Irregular triangle read by rows: T(n, k) is the number of factorizations of n into k factors (where 0 < k <= Omega(n)) that are inequivalent under toroidal symmetry.

1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 2, 1, 1, 4, 2, 1, 1, 2, 1, 2, 1, 4, 2, 3, 1, 1, 4, 2, 1, 1, 4, 2, 1, 2, 1, 2, 1, 1, 6, 6, 3, 1, 2, 1, 2, 1, 2, 2, 1, 4, 2, 1, 1, 6, 4, 1, 1, 4, 4, 3, 2, 1, 2, 1, 2, 1, 2, 1, 8, 8, 7, 1, 1, 2, 1, 2, 1, 6, 6, 3, 1, 1, 6, 4, 1, 1, 4, 2, 1, 4, 2, 1, 2, 1, 1, 8, 12, 12, 2, 1, 2, 1, 4, 2

Offset

2,4

Comments

Consult documentation link for detailed semantics. For a given n and k all tori obtained by attaching opposite sides of a rectangle with q rows and r columns and k = q*r slots are considered.

References

F. Harary and E. Palmer, Graphical Enumeration, Academic Press, 1973, page 37.

Links

Table of n, a(n) for n=2..109.

Marko Riedel, Counting toroidal factorizations.

Marko Riedel, Maple code to compute the sequence

Example

The triangle T(n, k) begins
  n\k| 1     2     3     4
  ---+--------------------
   2 | 1
   3 | 1
   4 | 1     2
   5 | 1
   6 | 1     2
   7 | 1
   8 | 1     2     2
   9 | 1     2
  10 | 1     2
  11 | 1
  12 | 1     4     2
  13 | 1
  14 | 1     2
  15 | 1     2
  16 | 1     4     2     3

Crossrefs

Cf. A335078, A335079, A392446.

Sequence in context: A359307 A205154 A337772 * A378444 A371243 A378284

Adjacent sequences: A392442 A392443 A392444 * A392446 A392447 A392448

Keyword

nonn,tabf

Author

Marko Riedel, Jan 12 2026

Status

approved