A074764 -id:A074764 - cp's OEIS frontend

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

Eric W. Weisstein

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.

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.

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).

Cf. A074764 (squares).

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

A005792 Positive numbers that are the sum of 2 squares or 3 times a square.

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 9, 10, 12, 13, 16, 17, 18, 20, 25, 26, 27, 29, 32, 34, 36, 37, 40, 41, 45, 48, 49, 50, 52, 53, 58, 61, 64, 65, 68, 72, 73, 74, 75, 80, 81, 82, 85, 89, 90, 97, 98, 100, 101, 104, 106, 108, 109, 113, 116, 117, 121, 122, 125, 128, 130, 136, 137, 144, 145

Offset: 1

N. J. A. Sloane

Equivalently, numbers of the form k^2, k^2+m^2, or 3*k^2, where k >= 1, m >= 1.

Theorem (Golomb; Snover et al.): A triangle can be partitioned into n pairwise congruent triangles iff n is of the form k^2, k^2+m^2, or 3*k^2.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
A. Soifer, How Does One Cut A Triangle?, Chapter 2, CEME, Colorado Springs CO 1990.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Solomon W. Golomb, Replicating figures in the plane, The mathematical gazette 48.366 (1964): 403-412.
Murray Klamkin, Review of "How Does One Cut a Triangle?" by Alexander Soifer, Amer. Math. Monthly, October 1991, pp. 775-. [Annotated scanned copy of pages 775-777 only] See "Grand Problem 1".
S. Snover, Letter to N. J. A. Sloane, May 1991
S. L. Snover, C. Wavereis and J. K. Williams, Rep-tiling for triangles, Discrete Math. 91 (1991), no. 2, 193-200.
Index entries for sequences related to sums of squares

Union of positive terms of A000290, A000404, A033428.

Cf. A074764.

More terms from Larry Reeves (larryr(AT)acm.org), Mar 21 2001

Entry revised by N. J. A. Sloane, Nov 30 2016

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

References

Links

Crossrefs

Extensions