A396562 a(n) is the minimal number of integer cubes needed to tile an integer cuboid of volume A033942(n).

1, 5, 2, 11, 6, 3, 1, 7, 16, 4, 10, 5, 21, 9, 12, 6, 22, 10, 2, 7, 20, 11, 1, 31, 12, 28, 9, 28, 13, 36, 10, 3, 14, 11, 12, 15, 5, 35, 14, 16, 46, 13, 25, 4, 40, 14, 51, 18, 13, 15, 19, 1, 15, 2, 46, 20, 5, 17, 61, 21, 18, 41, 22, 26, 19, 16, 49, 23, 6, 6, 24, 45

Offset

1,2

Comments

For volumes k not in A033942, any 3-factorization had at least one factor 1 and a(k) would be k.

First counterexample to guillotine optimality: a(43) = 25 for V = A033942(43) = 105 = 3*5*7. The best guillotine partition uses 25 cubes, while the true optimum is 32 and requires a nonguillotine arrangement.

Links

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

Felix Huber, Maple program

Formula

1 <= a(n) <= A033942(n) - A020639(A033942(n))^3 + 1.

a(n) = t for A033942(n) = t*m^3 and positive integers m and t.

a(n) = m^2 + 1 for A033942(n) = m^3 + m^2 and positive integer m.

a(n) = u*v for A033942(n) = t^3*u*v and positive integers t, u, v and u*v cubefree (A004709).

Example

a(43) = 25 for V = A033942(43) = 105 = 3*5*7. The optimal tiling uses two cubes of side 3, four cubes of side 2, and 19 unit cubes, for a total of 25 cubes:
  |   layer 1   |   layer 2   |   layer 3   |   layer 4   |   layer 5   |   layer 6   |   layer 7   |
  ---------------------------------------------------------------------------------------------------
  |    3 3 3    |    3 3 3    |    3 3 3    |    2 2 1    |    2 2 1    |    2 2 1    |    2 2 1    |
  |    3 3 3    |    3 3 3    |    3 3 3    |    2 2 1    |    2 2 1    |    2 2 1    |    2 2 1    |
  |    3 3 3    |    3 3 3    |    3 3 3    |    1 1 1    |    3 3 3    |    3 3 3    |    3 3 3    |
  |    2 2 1    |    2 2 1    |    2 2 1    |    2 2 1    |    3 3 3    |    3 3 3    |    3 3 3    |
  |    2 2 1    |    2 2 1    |    2 2 1    |    2 2 1    |    3 3 3    |    3 3 3    |    3 3 3    |

Maple

# See Huber link.

Crossrefs

Cf. A004709, A007913, A020639, A033942.

Sequence in context: A051308 A257327 A074642 * A194034 A163257 A332452

Adjacent sequences: A396559 A396560 A396561 * A396563 A396564 A396566

Keyword

nonn

Author

Felix Huber, Jun 01 2026

Status

approved