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cubic numbers
A cubic number is a figurate number of the form  $n^3$.

Every sufficiently large number can be written as the sum of at most 7 positive cubes. Nine cubes are needed only for 9 and 239 and eight cubes are needed for 15 numbers, the largest being 454.

It is not presently know if for all sufficiently large  $n$  less than 7 cubes are enough: 8042 is the largest known number which needs 7 cubes and Deshouillers et al. in 2000 conjectured that 7373170279850 is the largest integer that cannot be expressed as the sum of 4 nonnegative cubes.

Every multiple of 6 can be represented as a sum of 4 signed cubes, since

\[6n=(n+1)^3+(n-1)^3-n^3-n^3.\]

Mahler proved that 1 has infinitely many representations as 3 signed cubes.

Every cube is the difference between the squares of two consecutive triangular numbers  $n^3 = T_{n}^2-T_{n-1}^2$.

There is only one known palindromic cube whose base is not palindromic, i.e., 22013 = 10662526601.

The formula

\[n+\left\lfloor%
\sqrt[3]{n+\sqrt[3]{n}}
\right\rfloor
\]
by Lambek & Moser gives the  $n$-th non-cube.

The first cubic numbers are 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331, 1728, 2197, 2744, 3375, 4096, 4913, 5832, 6859, 8000 more terms

Cubic numbers can also be... (you may click on names or numbers and on + to get more values)

ABA 64 512 5832 + 9009299354112 9552418372232 aban 27 64 125 + 970299000000 1000000000000 abundant 216 1000 1728 + 49027896 49836032 alternating 27 125 216 + 47832147 381078125 amenable 64 125 216 + 994011992 1000000000 apocalyptic 3375 5832 6859 + 27000 29791 arithmetic 27 125 343 + 9663597 9938375 astonishing 27 216 c.heptagonal 1331 c.octagonal 729 15625 117649 + 940299110504209 976929722015625 c.triangular 64 cake 64 Canada 125 compositorial 1728 congruent 125 216 343 + 9800344 9938375 constructible 64 512 4096 + 35184372088832 281474976710656 Curzon 125 729 9261 + 154854153 161878625 d-powerful 5832 35937 287496 3944312 4657463 de Polignac 205379 226981 389017 + 64481201 91733851 decagonal 27 deficient 27 64 125 + 9800344 9938375 dig.balanced 216 2744 15625 + 179406144 187149248 droll 373248 13824000 303464448 + 191102976000000 809140700053504 Duffinian 27 64 125 + 9129329 9393931 eban 64 64000 64000000 64000000000 64000000000000 economical 27 64 125 + 19683000 19902511 enlightened 2744 262144 2097152 + 231928233984 373714754427 equidigital 27 64 1000 + 7762392 8489664 esthetic 343 evil 27 125 216 + 991026973 997002999 Friedman 125 216 343 + 941192 970299 frugal 125 343 512 + 994011992 997002999 gapful 1000 1331 1728 + 98611128000 99961946721 happy 1000 4096 12167 + 8869743 8998912 Harshad 27 216 512 + 9827847288 9910665792 Hogben 343 iban 27 343 1000 + 27000 343000 impolite 64 512 4096 + 35184372088832 281474976710656 inconsummate 216 9261 35937 + 658503 804357 interprime 64 1728 4096 + 90518849 91125000 Jordan-Polya 64 216 512 + 812479653347328 949978046398464 junction 216 4913 13824 + 98611128 99252847 katadrome 64 Leyland 512 lucky 729 29791 50653 + 7189057 9393931 Lynch-Bell 216 13824 magnanimous 512 metadrome 27 125 modest 578009537 1009027027 Moran 27 nialpdrome 64 1000 8000 + 8000000000000 64000000000000 nude 216 2744 13824 + 392223168 463684824 oban 27 512 729 odious 64 512 2197 + 994011992 1000000000 palindromic 343 1331 1030301 + 1030607060301 1334996994331 pancake 4096 pandigital 216 pernicious 2197 15625 17576 + 9393931 9663597 plaindrome 27 125 power 27 64 125 + 49994646057719 49998717504000 powerful 27 64 125 + 999940001199992 999970000299999 practical 64 216 512 + 9261000 9528128 pseudoperfect 216 1000 1728 + 941192 1000000 repunit 343 Rhonda 5832 15625 Ruth-Aaron 125 1331 10648 148877000 self 64 512 1728 + 967361669 973242271 Smith 27 729 19683 + 8489664 62099136 square 64 729 4096 + 976929722015625 995686217814016 star 226981 super Niven 1000 8000 1000000 + 1000000000 8000000000 super-d 1331 3375 6859 + 9261000 9393931 tau 21952 32768 64000 + 976191488 1000000000 trimorphic 125 uban 27 1000000 8000000 + 16003008000000 27000000000000 Ulam 19683 97336 175616 + 7077888 8000000 undulating 343 unprimeable 512 10648 21952 + 9528128 9800344 untouchable 216 1728 5832 + 884736 941192 upside-down 64 vampire 26198073 wasteful 216 27000 74088 + 474552 9261000 Zuckerman 216 13824 4741632 Zumkeller 216 1000 1728 + 74088 85184