Brazilian numbers are so called as they were first formally presented at the 1994 math Olympiad Olimpiada Iberoamericana de Matematica in Fortaleza, Brazil.

Brazilian numbers are defined as:

The set of positive integer numbers where each number N has at least one natural number B where 1 < B < N-1 where the representation of N in base B has all equal digits.



E.G.

 1, 2 & 3 can not be Brazilian; there is no base B that satisfies the condition 1 < B < N-1.
 4 is not Brazilian; 4 in base 2 is 100. The digits are not all the same.
 5 is not Brazilian; 5 in base 2 is 101, in base 3 is 12. There is no representation where the digits are the same.
 6 is not Brazilian; 6 in base 2 is 110, in base 3 is 20, in base 4 is 12. There is no representation where the digits are the same.
 7 is Brazilian; 7 in base 2 is 111. There is at least one representation where the digits are all the same.
 8 is Brazilian; 8 in base 3 is 22. There is at least one representation where the digits are all the same.
 and so on...


All even integers 2P >= 8 are Brazilian because 2P = 2(P-1) + 2, which is 22 in base P-1 when P-1 > 2. That becomes true when P >= 4.

More common: for all all integers R and S, where R > 1 and also S-1 > R, then R*S is Brazilian because R*S = R(S-1) + R, which is RR in base S-1

The only problematic numbers are squares of primes, where R = S. Only 11^2 is brazilian to base 3.

All prime integers, that are brazilian, can only have the digit 1. Otherwise one could factor out the digit, therefore it cannot be a prime number. Mostly in form of 111 to base Integer(sqrt(prime number)). Must be an odd count of 1 to stay odd like primes > 2


Task

Write a routine (function, whatever) to determine if a number is Brazilian and use the routine to show here, on this page;

 the first 20 Brazilian numbers;
 the first 20 odd Brazilian numbers;
 the first 20 prime Brazilian numbers;


See also

 OEIS:A125134 - Brazilian numbers
 OEIS:A257521 - Odd Brazilian numbers
 OEIS:A085104 - Prime Brazilian numbers


 
   Doc=: conjunction def 'u :: (n"_)'
 
   brazilian=: (1 e. (#@~.@(#.^:_1&>)~ (2 + [: (i.) _3&+)))&> Doc 'brazilian y  NB. is 1 if y is brazilian, else 0'
 
   Filter=: (#~`)(`:6)
 
 
   B=: brazilian Filter 4 + i. 300   NB. gather Brazilion numbers less than 304
 
   20 {. B   NB. first 20 Brazilion numbers
7 8 10 12 13 14 15 16 18 20 21 22 24 26 27 28 30 31 32 33
 
   odd =: 1 = 2&|
 
   20 {. odd Filter B   NB. first 20 odd Brazilion numbers
7 13 15 21 27 31 33 35 39 43 45 51 55 57 63 65 69 73 75 77
 
   prime=: 1&p:
 
   20 {. prime Filter B   NB. uh oh need a new technique
7 13 31 43 73 127 157 211 241 0 0 0 0 0 0 0 0 0 0 0
 
   NB. p: y   is the yth prime, with 2 being prime 0
   20 {. brazilian Filter p: 2 + i. 500
7 13 31 43 73 127 157 211 241 307 421 463 601 757 1093 1123 1483 1723 2551 2801
 