This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A014544 #41 Feb 16 2025 08:32:33 %S A014544 1,8,15,20,22,27,29,34,36,38,39,41,43,45,46,48,49,50,51,52,53,54,55, %T A014544 56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78, %U A014544 79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101 %N A014544 Numbers k such that a cube can be divided into k subcubes. %C A014544 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. %C A014544 Combining these facts gives the remaining terms shown and all numbers > 47. %C A014544 It may or may not have been shown that no other numbers occur - see Hickerson link. %D A014544 J.-P. Delahaye, Les inattendus mathématiques, p. 93, Belin-Pour la science, Paris, 2004. %D A014544 Howard Eves, A Survey of Geometry, Vol. 1. Allyn and Bacon, Inc., Boston, Mass. 1966, see p. 271. %D A014544 M. Gardner, Fractal Music, Hypercards and More: Mathematical Recreations from Scientific American Magazine. New York: W. H. Freeman, pp. 297-298, 1992. %H A014544 Peter Connor and Phillip Marmorino, <a href="https://doi.org/10.1007/s00022-018-0424-4">Decomposing cubes into smaller cubes</a>, Journal of Geometry 109 (2018), article 19. %H A014544 Dean Hickerson, <a href="/A014544/a014544.txt">Further comments on A014544</a>, Nov 01 2007 and Nov 10 2007 %H A014544 Matthew Hudelson, <a href="https://doi.org/10.1006/jcta.1997.2837">Dissecting d-cubes into smaller d-cubes</a>, Journal of Combinatorial Theory, Series A, 81 (1998), 190-200. %H A014544 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CubeDissection.html">Cube Dissection</a>. %H A014544 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HadwigerProblem.html">Hadwiger Problem</a>. %H A014544 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1). %Y A014544 Cf. A074764 (squares). %K A014544 easy,nonn %O A014544 1,2 %A A014544 _Eric W. Weisstein_ %E A014544 More terms from _Jud McCranie_, Mar 19 2001, who remarks that all integers > 47 are in the sequence. %E A014544 Edited by _Dean Hickerson_, Jan 05 2003