Editing Repdigit (section)

==''k''-Brazilian numbers==
* The number of ways such that a number ''n'' is Brazilian is in {{OEIS2C|id=A220136}}. Hence, there exist numbers that are non-Brazilian and others that are Brazilian; among these last integers, some are once Brazilian, others are twice Brazilian, or three times, or more. A number that is ''k'' times Brazilian is called ''k-Brazilian number''.
*Non-Brazilian numbers or 0''-Brazilian numbers'' are constituted with 1 and 6, together with some primes and some squares of primes. The sequence of the non-Brazilian numbers begins with 1, 2, 3, 4, 5, 6, 9, 11, 17, 19, 23, 25, ... {{OEIS|id=A220570}}.
* The sequence of 1''-Brazilian numbers'' is composed of other primes, the only square of prime that is Brazilian, 121, and composite numbers {{nowrap|≥ 8}} that are the product of only two distinct factors such that {{nowrap|1=''n'' = ''a'' × ''b'' = ''aa''<sub>''b''–1</sub>}} with {{nowrap|1 < ''a'' < ''b'' – 1}}.  {{OEIS|id=A288783}}.
* The 2''-Brazilian numbers'' {{OEIS|id=A290015}} consists of composites and only two primes: 31 and 8191. Indeed, according to [[Goormaghtigh conjecture]], these two primes are the only known solutions of the [[Diophantine equation]]: <div style="text-align: center;"><math>p=\frac{x^m - 1}{x-1}=\frac{y^n - 1}{y - 1}</math> with ''x'', ''y''&nbsp;>&nbsp;1 and ''n'', ''m''&nbsp;>&nbsp;2 :</div>  
**(''p'',&nbsp;''x'',&nbsp;''y'',&nbsp;''m'',&nbsp;''n'')&nbsp;=&nbsp;(31,&nbsp;5,&nbsp;2,&nbsp;3,&nbsp;5) corresponding to 31 = 11111<sub>2</sub> = 111<sub>5</sub>, and,
**(''p'',&nbsp;''x'',&nbsp;''y'',&nbsp;''m'',&nbsp;''n'')&nbsp;=&nbsp;(8191,&nbsp;90,&nbsp;2,&nbsp;3,&nbsp;13) corresponding to 8191 = 1111111111111<sub>2</sub> = 111<sub>90</sub>, with 11111111111 is the [[repunit]] with thirteen digits 1.
* For each sequence of ''k-Brazilian numbers'', there exists a smallest term. The sequence with these smallest ''k''-Brazilian numbers begins with 1, 7, 15, 24, 40, 60, 144, 120, 180, 336, 420, 360, ... and are in {{OEIS2C|id=A284758}}. For instance, 40 is the smallest ''4-Brazilian number'' with  40 = 1111<sub>3</sub> = 55<sub>7</sub> = 44<sub>9</sub> = 22<sub>19</sub>.
* In the ''Dictionnaire de (presque) tous les nombres entiers'',<ref>{{cite book|title=Dictionnaire de (presque) tous les nombres entiers  |author=Daniel Lignon|location=Paris|publisher=Ellipses|pages=420<!--endif p.totales-->|year=2012}}</ref> Daniel Lignon proposes that an integer is ''highly Brazilian'' if it is a positive integer with more Brazilian representations  than any smaller positive integer has. This definition comes from the definition of [[highly composite number]]s created by [[Srinivasa Ramanujan]] in 1915. The first numbers ''highly Brazilian'' are  1, 7, 15, 24, 40, 60, 120, 180, 336, 360, 720, ... and are exactly in {{OEIS2C|id=A329383}}. From 360 to 321253732800 (maybe more), there are 80 successive [[highly composite number]]s that are also highly Brazilian numbers, see  {{OEIS2C|id=A279930}}.