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A014544
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Numbers n such that a cube can be divided into n sub-cubes.
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3
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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
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OFFSET
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1,2
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COMMENTS
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If m and n are in the sequence, so is m+n-1, since n-dissecting one cube in an m-dissection gives an (m+n-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.
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REFERENCES
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J.-P. Delahaye, Les inattendus mathematiques, pp. 93 Belin-Pour la science, Paris, 2004.
Eves, Howard, 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.
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LINKS
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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 to sequences with linear recurrences with constant coefficients, signature (2,-1).
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CROSSREFS
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Sequence in context: A160524 A161541 A133157 * A122754 A082867 A075713
Adjacent sequences: A014541 A014542 A014543 * A014545 A014546 A014547
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KEYWORD
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easy,nonn
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AUTHOR
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Eric W. Weisstein
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EXTENSIONS
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More terms from Jud McCranie, Mar 19 2001, who remarks that all integers > 47 are in the sequence.
Edited by Dean Hickerson (dean.hickerson(AT)yahoo.com), Jan 05 2003
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STATUS
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approved
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