Cubic Number

Author

Eric W. Weisstein
January 18, 2009

This notebook downloaded from http://mathworld.wolfram.com/notebooks/IntegerSequences/CubicNumber.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/CubicNumber.html.

©2009 Wolfram Research, Inc. except for portions noted otherwise

Binary Plot

<<Utilities`Plot`

Show[PixelPerfect[BasePlot[CubicNumber/@Range[255]],BlockSize->2]]

Show[PixelPerfect[BasePlot[CubicNumber/@Range[511]],BlockSize->1]]

Enumeration

Range[20]^3

Generating Function

<<DiscreteMath`RSolve`

PowerSum[n^3,{x,n}]

All numbers not of the form 9 n±4 a requiring 3 cubes

http://www.asahi-net.or.jp/~KC2H-MSM/mathland/math04/matb0100.htm
http://www.asahi-net.or.jp/~KC2H-MSM/mathland/math04/cube00.htm

l={
{30,-283059965,-2218888517,2220422932},
{52,60702901317,23961292454,-61922712865},
{75,435203083,-435203231,4381159},
{84,41639611,-41531726,-8241191},
{110,109938919,16540290030,-16540291649},{156,68844645625,2232194323,-68845427846},
{195,-2238006277,-5087472163,5227922915},
{290,426417007,2070897315,-2076906362},
{318,47835963799,20549442727,-49068024704},{366,241832223257,167734571306,-266193616507},{420,8859060149051,-2680209928162,-8776520527687},{435,4460467,-4078175,-2755337},
{444,3460795,14820289,-14882930},
{452,-2267462975,-3041790413,3414300774},
{462,1933609,-1832411,-1024946},
{478,-1368722,-13434503,13439237},
{501,29500376,-19895059,-26109316},
{530,45623198,-11079901,-45404321},
{564,53872419107,-1300749634,-53872166335},{588,5657478787,-3650204951,-5097345554},
{600,870406166,8762318,-870406462},
{606,2638615657,761302981,-2659573862},
{609,30430569722,20129024420,-33121417879},
{618,10500707,-6956785,-9364360},
{735,1076499359,815238200,-1214029864},{758,662325744409,109962567936,-663334553003},{786,21050373397,12791387317,-22519801850},{789,18918117957926,4836228687485,-19022888796058},{830,143271714,125723125,-170174279},
{834,47272468418,25585441403,-49649244505},{894,19868127639556,2322626411251,-19878702430997},{948,323019573172,63657228055,-323841549995}};

{First[#],First[#]-Total[Rest[#]^3]}&/@l//Column

Cubes of Rationals sum to integers

Plus@@({17/21 ,37/21}^3)

Plus@@({415280564497/348671682660,676702467503/ 348671682660}^3)