Brazilian numbers

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This page is the answer to the task Brazilian numbers in the Rosetta Code.

Description (from Rosetta Code)

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: all integers, that factor decomposition is R*S >= 8, with S+1 > R, are 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

Program

The following function calculates whether a given number is Brazilian:

BrazilianProgram1.png

We can use the following function to list the first Brazilian numbers, according with a given condition, passed as a lambda expression:

BrazilianProgram2.png

Case 1. The first 20 Brazilian numbers

BrazilianCase1.png

Case 2. The first 20 odd Brazilian numbers

BrazilianCase2.png

Case 3. The first 20 prime Brazilian numbers

BrazilianCase3.png