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A288783 Brazilian numbers which have only one Brazilian representation. 4

%I

%S 7,8,10,12,13,14,16,20,22,27,33,34,35,38,39,43,46,51,55,58,65,69,73,

%T 74,77,81,82,87,94,95,106,115,118,119,121,122,123,125,127,134,141,142,

%U 143,145

%N Brazilian numbers which have only one Brazilian representation.

%C These numbers could be called 1-Brazilian numbers.

%C The smallest number of this sequence is 7 = 111_2 which is also the smallest Brazilian number (A125134) and so the smallest Brazilian prime (A085104).

%C a(2) = 8 is the smallest composite Brazilian number and so the smallest even composite Brazilian with 8 = 22_3 (A220571).

%C a(10) = 27 is the smallest odd composite Brazilian in this sequence because 27 = 33_8 but 15 is the smallest odd composite Brazilian with 15 = 1111_2 = 33_4 so with two representations.

%C 121 is the only square of prime which is Brazilian with 121 = 11111_3.

%C In this sequence, there are:

%C 1) The Brazilian primes and the lone square prime 121 which are all repunits in a base >= 2 with a string of at least three 1's.

%C 2) The composite numbers which are such that n = a * b = (aa)_(b-1) with 1 < a < b-1 with only one such product a * b.

%D D. Lignon, Dictionnaire de (presque) tous les nombres entiers, Ellipses, 2012, page 420.

%H Bernard Schott, <a href="/A125134/a125134.pdf">Les nombres brésiliens</a>, Quadrature, no. 76, avril-juin 2010, pages 30-38.

%e 13 = 111_3; 127 = 1111111_2.

%e 20 = 2 * 10 = 22_9; 55 = 5 * 11 = 55_10; 69 = 3 * 23 = 33_22.

%t Select[Range@ 145, Function[n, Count[Range[2, n - 2], b_ /; SameQ @@ IntegerDigits[n, b]] == 1]] (* _Michael De Vlieger_, Jun 16 2017 *)

%Y Cf. A085104, A125134, A190300, A220570, A220571, A253260, A284758.

%K nonn,base,easy

%O 1,1

%A _Bernard Schott_, Jun 15 2017

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Last modified February 21 23:21 EST 2020. Contains 332113 sequences. (Running on oeis4.)