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Extraordinary cubes and roots
(non-palindromic allowed)
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rood Extra squares rood


Introduction

Palindromic numbers are numbers which read the same from
 p_right left to right (forwards) as from the right to left (backwards) p_left
Here are a few random examples : 9, 9559, 4078704, 9876543210123456789


Extraordinary (Palindromic) Cubes and Cuberoots

  Topic 2  
Cubes containing their cuberoots in base 10
by Patrick De Geest [ November 16, 2025 ]

OEIS A029942 Numbers k such that the decimal expansion of k^3 contains k as a substring.
OEIS A052210NOT IN OEIS   OEIS A383640NOT IN OEIS   OEIS A033819OEIS A215558
Root LEFTCube LEFT   Root MIDDLECube MIDDLE   Root RIGHTCube RIGHT
\(0\\1\\10\\32\\100\\1000\\10000\\31623\\100000\\316228\\\cdots \)\({\color{royalblue}{0}}\\{\color{royalblue}{1}}\\{\color{royalblue}{10}}00\\{\color{royalblue}{32}}768\\{\color{royalblue}{100}}0000\\{\color{royalblue}{1000}}000000\\{\color{royalblue}{10000}}00000000\\{\color{royalblue}{31623}}446801367\\{\color{royalblue}{100000}}0000000000\\{\color{royalblue}{316228}}46796684352\\\cdots \)   \(56\\782\\5111\\8089\\8216\\9553\\11768\\14357\\18229\\53257\\\cdots \)\(17{\color{royalblue}{56}}16\\4{\color{royalblue}{782}}11768\\133{\color{royalblue}{5111}}82631\\529278{\color{royalblue}{8089}}69\\554601{\color{royalblue}{8216}}96\\871804{\color{royalblue}{9553}}77\\162970{\color{royalblue}{11768}}32\\29593{\color{royalblue}{14357}}293\\605743{\color{royalblue}{18229}}89\\1510{\color{royalblue}{53257}}765593\\\cdots \)   \(0\\1\\4\\5\\6\\9\\24\\25\\49\\51\\\cdots \)\({\color{royalblue}{0}}\\{\color{royalblue}{1}}\\6{\color{royalblue}{4}}\\12{\color{royalblue}{5}}\\21{\color{royalblue}{6}}\\72{\color{royalblue}{9}}\\138{\color{royalblue}{24}}\\156{\color{royalblue}{25}}\\1176{\color{royalblue}{49}}\\1326{\color{royalblue}{51}}\\\cdots \) 
\\ Some Pari/gp code by P. De Geest
\\ Swap the second line with one of the bottom three lines if you want subsets only.
{
e=3; for(p=0,10000000, pr=digits(p); pp=digits(p^e); prl=#pr; q=#pp-prl+1;
for(i=1,q, if(pr==pp[i..i+prl-1], print(i,"  ",pr," ",pp))));
}

/*
for(i=1,q, if(pr==pp[i..i+prl-1], print(i,"  ",pr," ",pp)))); \\ EVERYWHERE

for(i=1,q, if(pr==pp[i..i+prl-1]&&i==1, print(i,"  ",pr," ",pp)))); \\ AT LEFT

for(i=1,q, if(pr==pp[i..i+prl-1]&&i>1&&i<q, print(i,"  ",pr," ",pp)))); \\ AT MIDDLE

for(i=1,q, if(pr==pp[i..i+prl-1]&&i==q, print(i,"  ",pr," ",pp)))); \\ AT RIGHT
*/

Further reading : “Exploring the Beauty of Fascinating Numbers”, Shyam Sunder Gupta, Springer, 2025, p.59 [2.9.7]



  Topic 1  
Exclusionary Squares and Cubes

E.g.:
2038792 = 41566646641
6391722 = 408540845584

63783 = 259449922152
76583 = 449103134312

Note: this topic has been treated in depth at my webpage ninedig2.htm#topic2.8

Here is an interesting webpage from the collection of Clif Pickover
Clif Pickover's Extreme Challenges in Mathematics and Morals
Exclusionary Squares and Cubes

Here are some corresponding OEIS entries

With no repeating digits in the base numbers
A112321 Least n-digit number such that its square is exclusionary, or 0 if no such number exists. - Lekraj Beedassy
A112322 Exclusionary square associated to corresponding smallest n-digit number (A112321), or 0 if no such number exists. - Lekraj Beedassy and Klaus Brockhaus
A113951 Largest number whose n-th power is exclusionary (or 0 if no such number exists). - Lekraj Beedassy
A113952 Largest exclusionary n-th power (or 0 if no such number exists). - Lekraj Beedassy
A112993 Exclusionary cubes: cubes of the terms in A112994. - Lekraj Beedassy
A112994 Numbers whose cubes are exclusionary: numbers n such that n and n^3 have no digits in common. - Lekraj Beedassy

With repeating digits in the base numbers
A029783 Exclusionary squares: numbers n such that digits of n are not present in n^2. - Patrick De Geest
A029784 Squares such that digits of sqrt(n) are not present in n. - Patrick De Geest
A029785 Digits of n are not present in n^3. - Patrick De Geest
A029786 Cubes such that digits of cube root of n are not present in n. - Patrick De Geest

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( © All rights reserved ) - Last modified : November 16, 2025.
Patrick De Geest - Belgium flag - Short Bio - Some Pictures
E-mail address : pdg@worldofnumbers.com