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trimorphic numbers
A number  $n$  is called trimorphic if  $n^3$  ends with  $n$.

For example,  $99$  is trimorphic since  $99^3 = 9702\underline{99}$.

The first trimorphic numbers are 1, 4, 5, 6, 9, 24, 25, 49, 51, 75, 76, 99, 125, 249, 251, 375, 376, 499, 501, 624, 625, 749, 751, 875, 999 more terms

Pictorial representation of remainders (mod 2, 3, ...,11) frequency. For a table of values and more details click here
remainders of trimorphic numbers

A graph displaying how many trimorphic numbers are multiples of the primes p from 2 to 71. In black the ideal line 1/p.
divisibility of trimorphic numbers
Useful links Mathworld, Trimorphic Number
Wikipedia, Trimorphic number
OEIS, Sequence A033819
Trimorphic numbers can also be... (you may click on names or numbers and on + to get more values)

a-pointer 574218751 ABA 24 375 aban 24 25 49 + 500000000001 abundant 24 624 90624 + 12890624 Achilles 2109375 admirable 24 alternating 25 49 76 + 749 amenable 24 25 49 + 925781249 apocalyptic 251 499 624 + 18751 arithmetic 49 51 99 + 9999999 automorphic 25 76 376 + 740081787109376 betrothed 75 brilliant 25 49 c.octagonal 25 49 625 + 8212890625 c.pentagonal 51 76 c.square 25 Canada 125 Chen 251 499 751 + 281249 compositorial 24 congruent 24 125 375 + 9999999 constructible 24 51 cube 125 Cullen 25 Cunningham 24 99 624 + 99999999999999 Curzon 125 249 749 + 25781249 cyclic 51 249 251 + 4999999 D-number 51 249 501 + 500001 d-powerful 24 375 376 + 59375 de Polignac 251 499999 7890625 87109375 decagonal 3751 deficient 25 49 51 + 9999999 dig.balanced 49 75 99 + 50000001 Duffinian 25 49 75 + 7890625 economical 25 49 125 + 12890625 emirp 751 1249 emirpimes 49 51 18751 equidigital 25 49 249 + 12890624 eRAP 24 esthetic 76 evil 24 51 75 + 925781249 factorial 24 fibodiv 75 Friedman 25 125 625 + 781249 frugal 125 625 4375 + 787109375 gapful 3751 9999 109376 + 36425781249 Gilda 49 good prime 251 happy 49 376 8751 109376 Harshad 24 375 624 + 999999999 highly composite 24 iban 24 idoneal 24 25 inconsummate 75 9375 59375 + 281249 interprime 76 99 376 + 7109376 Jordan-Polya 24 junction 5625 90625 890625 2890625 Kaprekar 99 999 9999 + 999999999999999 katadrome 51 75 76 + 8751 Lehmer 51 Lucas 76 lucky 25 49 51 + 9218751 Lynch-Bell 24 624 magnanimous 25 49 76 + 625 metadrome 24 25 49 + 1249 modest 49 499 999 + 999999999 Moran 999 Motzkin 51 nialpdrome 51 75 76 + 999999999999999 nonagonal 24 75 nude 24 99 624 + 99999999 oban 25 75 76 + 999 odious 25 49 76 + 999999999 palindromic 99 999 9999 + 999999999999999 panconsummate 24 pandigital 75 99 pentagonal 51 376 pernicious 24 25 49 + 7890625 Perrin 51 plaindrome 24 25 49 + 999999999999999 power 25 49 125 + 8212890625 powerful 25 49 125 + 8212890625 practical 24 624 90624 + 2890624 prim.abundant 999999 prime 251 499 751 + 574218751 Proth 25 49 90625 pseudoperfect 24 624 90624 + 999999 repdigit 99 999 9999 + 999999999999999 repfigit 75 Ruth-Aaron 24 25 49 125 self 75 501 749 + 425781249 semiprime 25 49 51 + 500001 sliding 25 Smith 7890625 Sophie Germain 251 74218751 sphenic 50001 781249 5000001 + 75781249 square 25 49 625 + 8212890625 straight-line 999 9999 99999 + 999999999999999 strong prime 251 499 751 + 4999999 super Niven 24 super-d 749 781249 890624 superabundant 24 tau 24 376 625 + 2890624 tribonacci 24 twin 31249 281249 uban 25 49 51 + 50000000000001 Ulam 99 624 751 + 9999999 unprimeable 624 625 90624 + 7890625 untouchable 624 9376 890624 wasteful 24 51 75 + 9999999 weak prime 31249 49999 74218751 Woodall 99999999999 Zuckerman 24 624 Zumkeller 24 624 90624 zygodrome 99 999 9999 + 999999999999999