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.

7, 8, 10, 12, 13, 14, 15, 16, 18, 20, 21, 22, 24, 26, 27, 28, 30, 31, 32, 33, 34, 35, 36, 38, 39, 40, 42, 43, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 73, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90

Offset: 1

Author

Bernard Schott, Jan 21 2007

Keywords

• nonn
• base
• easy

Comments

The condition b < n-1 is important because every number n has representation 11 in base n-1. - Daniel Lignon, May 22 2015

Every even number >= 8 is Brazilian. Odd Brazilian numbers are in A257521. - Daniel Lignon, May 22 2015

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

Examples

			15 is a member since it is 33 in base 4.
		

References

• Pierre Bornsztein, "Hypermath", Vuibert, Exercise a35, p. 7.

Links

• Nathaniel Johnston, Table of n, a(n) for n = 1..4000
• 9th Iberoamerican Mathematical Olympiad, Problem 1: sensible numbers, Fortaleza, Brazil, September 17-25, 1994.
• Bernard Schott and others, Postings to the French mathematical forum les-mathematiques.net
• Bernard Schott, Les nombres brésiliens, Quadrature, no. 76, avril-juin 2010, pages 30-38; included here with permission from the editors of Quadrature.

Crossrefs

Cf. A190300 and A257521 (odd Brazilian numbers).

Cf. A085104 (prime Brazilian numbers).

Programs

Maple

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

Mathematica

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 *)

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

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

Formula

a(n) ~ n. - Charles R Greathouse IV, Aug 09 2017

Extensions

More terms from Nathaniel Johnston, May 24 2011