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

1,2

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

Table of n, a(n) for n=1..69.

Dean Hickerson, Further comments on A014544, Nov 01 2007 and Nov 10 2007

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

Sequence in context: A161541 A247081 A133157 * A237610 A122754 A355490

Adjacent sequences: A014541 A014542 A014543 * A014545 A014546 A014547

Keyword

easy,nonn

Author

Eric W. Weisstein

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