A014544 Numbers k such that a cube can be divided into k subcubes.

1, 8, 15, 20, 22, 27, 29, 34, 36, 38, 39, 41, 43, 45, 46, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101

Offset: 1

Author

Eric W. Weisstein

Keywords

• easy
• nonn

Comments

If m and j are in the sequence, so is m+j-1, since j-dissecting one cube in an m-dissection gives an (m+j-1)-dissection. 1, 8, 20, 38, 49, 51, 54 are in the sequence because of dissections corresponding to the equations 1^3 = 1^3, 2^3 = 8*1^3, 3^3 = 2^3 + 19*1^3, 4^3 = 3^3 + 37*1^3, 6^3 = 4*3^3 + 9*2^3 + 36*1^3, 6^3 = 5*3^3 + 5*2^3 + 41*1^3 and 8^3 = 6*4^3 + 2*3^3 + 4*2^3 + 42*1^3.

Combining these facts gives the remaining terms shown and all numbers > 47.

It may or may not have been shown that no other numbers occur - see Hickerson link.

References

• J.-P. Delahaye, Les inattendus mathématiques, p. 93, Belin-Pour la science, Paris, 2004.
• Howard Eves, A Survey of Geometry, Vol. 1. Allyn and Bacon, Inc., Boston, Mass. 1966, see p. 271.
• M. Gardner, Fractal Music, Hypercards and More: Mathematical Recreations from Scientific American Magazine. New York: W. H. Freeman, pp. 297-298, 1992.

Links

• Peter Connor and Phillip Marmorino, Decomposing cubes into smaller cubes, Journal of Geometry 109 (2018), article 19.
• Dean Hickerson, Further comments on A014544, Nov 01 2007 and Nov 10 2007
• Matthew Hudelson, Dissecting d-cubes into smaller d-cubes, Journal of Combinatorial Theory, Series A, 81 (1998), 190-200.
• Eric Weisstein's World of Mathematics, Cube Dissection.
• Eric Weisstein's World of Mathematics, Hadwiger Problem.
• Index entries for linear recurrences with constant coefficients, signature (2,-1).

Crossrefs

Cf. A074764 (squares).

Extensions

More terms from Jud McCranie, Mar 19 2001, who remarks that all integers > 47 are in the sequence.

Edited by Dean Hickerson, Jan 05 2003