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A125134 "Brazilian" numbers ("les nombres brésiliens" in French): numbers n such that there is a natural number b with 1 < b < n-1 such that the representation of n in base b has all equal digits.

%I A125134 #77 Nov 09 2018 18:25:36
%S A125134 7,8,10,12,13,14,15,16,18,20,21,22,24,26,27,28,30,31,32,33,34,35,36,
%T A125134 38,39,40,42,43,44,45,46,48,50,51,52,54,55,56,57,58,60,62,63,64,65,66,
%U A125134 68,69,70,72,73,74,75,76,77,78,80,81,82,84,85,86,87,88,90
%N A125134 "Brazilian" numbers ("les nombres brésiliens" in French): numbers n such that there is a natural number b with 1 < b < n-1 such that the representation of n in base b has all equal digits.
%C A125134 The condition b < n-1 is important because every number n has representation 11 in base n-1. - _Daniel Lignon_, May 22 2015
%C A125134 Every even number >= 8 is Brazilian. Odd Brazilian numbers are in A257521. - _Daniel Lignon_, May 22 2015
%C A125134 Looking at A190300, it seems that asymptotically 100% of composite numbers are Brazilian, while looking at A085104, it seems that asymptotically 0% of prime numbers are Brazilian. The asymptotic density of Brazilian numbers would thus be 100%. - _Daniel Forgues_, Oct 07 2016
%D A125134 Pierre Bornsztein, "Hypermath", Vuibert, Exercise a35, p. 7.
%H A125134 Nathaniel Johnston, <a href="/A125134/b125134.txt">Table of n, a(n) for n = 1..4000</a>
%H A125134 9th Iberoamerican Mathematical Olympiad, <a href="http://imomath.com/othercomp/Ib/IbMO94.pdf">Problem 1: sensible numbers</a>, Fortaleza, Brazil, September 17-25, 1994.
%H A125134 Bernard Schott and others, <a href="http://www.les-mathematiques.net/phorum/read.php?5,350062,page=1">Postings to the French mathematical forum les-mathematiques.net</a>
%H A125134 Bernard Schott, <a href="/A125134/a125134.pdf">Les nombres brésiliens</a>, Quadrature, no. 76, avril-juin 2010, pages 30-38; included here with permission from the editors of Quadrature.
%F A125134 a(n) ~ n. - _Charles R Greathouse IV_, Aug 09 2017
%e A125134 15 is a member since it is 33 in base 4.
%p A125134 isA125134 := proc(n) local k: for k from 2 to n-2 do if(nops(convert(convert(n,base,k),set))=1)then return true: fi: od: return false: end: A125134 := proc(n) option remember: local k: if(n=1)then return 7: fi: for k from procname(n-1)+1 do if(isA125134(k))then return k: fi: od: end: seq(A125134(n),n=1..65); # _Nathaniel Johnston_, May 24 2011
%t A125134 fQ[n_] := Module[{b = 2, found = False}, While[b < n - 1 && Length[Union[IntegerDigits[n, b]]] > 1, b++]; b < n - 1]; Select[Range[4, 90], fQ] (* _T. D. Noe_, May 07 2013 *)
%o A125134 (PARI) for(n=4,100,for(b=2,n-2,d=digits(n,b);if(vecmin(d)==vecmax(d),print1(n,", ");break))) \\ _Derek Orr_, Apr 30 2015
%o A125134 (PARI) is(n)=my(m); if(!isprime(n), return(if(issquare(n,&m), m>3 && (!isprime(m) || m==11), n>6))); for(b=2,n-2, m=digits(n,b); for(i=2,#m, if(m[i]!=m[i-1], next(2))); return(1)); 0 \\ _Charles R Greathouse IV_, Aug 09 2017
%Y A125134 Cf. A190300 and A257521 (odd Brazilian numbers).
%Y A125134 Cf. A085104 (prime Brazilian numbers).
%K A125134 nonn,base,easy
%O A125134 1,1
%A A125134 _Bernard Schott_, Jan 21 2007
%E A125134 More terms from _Nathaniel Johnston_, May 24 2011