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A014544
Numbers k such that a cube can be divided into k subcubes.
5
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
1,2
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.
CROSSREFS
Cf. A074764 (squares).
Sequence in context: A161541 A247081 A133157 * A237610 A122754 A355490
KEYWORD
easy,nonn
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
STATUS
approved