A380455 Maximum number of prime polyomino factors of an n-polyomino.

1, 1, 1, 1, 3, 1, 4, 1, 5, 1, 5, 1

Offset

2,5

Comments

Grade one students are capable of understanding "prime" and "composite" and "is a factor of" when presented in terms of polyomino tilings. Exploring these ideas is a 10/10 classroom activity, even if limited to only pentomino factors of larger polyominoes.

Conjectured terms: a(14)..a(17) = 10?, 3, 12?, 1.

Links

Table of n, a(n) for n=2..13.

Formula

a(p) = 1 if p is prime. - Pontus von Brömssen, Jun 24 2025

Example

a(6) = 3 because the 2x3 rectangular hexomino can be tiled by three prime polyominoes:
The domino:
  XOY
  XOY
The bent tromino:
  XXO
  XOO
The straight tromino:
  XXX
  OOO
a(9) = 1 because no 9-polyomino can be tiled by both all straight and all bent trominoes.
It is conjectured that a(14) = 10 because this 14-polyomino can be tiled with 9 prime heptominoes and by the domino:
    X
   XXXX
   XXXX
   XXXX
     X
It is also conjectured that a(16) = 12 because this 16-polyomino can be tiled with twelve prime 8-polyominoes:
     XXX
    XXXXX
   XXXXX
    XXX

Crossrefs

Cf. A342430 (number of prime polyominoes with n cells).

Sequence in context: A349042 A375820 A302792 * A179820 A364098 A363521

Adjacent sequences: A380452 A380453 A380454 * A380456 A380457 A380458

Keyword

nonn,more

Author

Gordon Hamilton, Jun 22 2025

Status

approved