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A125134
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"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.
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64
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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
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OFFSET
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1,1
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COMMENTS
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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
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REFERENCES
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Pierre Bornsztein, "Hypermath", Vuibert, Exercise a35, p. 7.
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LINKS
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Bernard Schott, Les nombres brésiliens, Quadrature, no. 76, avril-juin 2010, pages 30-38; included here with permission from the editors of Quadrature.
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FORMULA
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EXAMPLE
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15 is a member since it is 33 in base 4.
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MAPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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Cf. A085104 (prime Brazilian numbers).
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KEYWORD
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nonn,base,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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