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A288783
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Brazilian numbers which have only one Brazilian representation.
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5
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7, 8, 10, 12, 13, 14, 16, 20, 22, 27, 33, 34, 35, 38, 39, 43, 46, 51, 55, 58, 65, 69, 73, 74, 77, 81, 82, 87, 94, 95, 106, 115, 118, 119, 121, 122, 123, 125, 127, 134, 141, 142, 143, 145
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OFFSET
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1,1
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COMMENTS
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These numbers could be called 1-Brazilian numbers.
The smallest number of this sequence is 7 = 111_2 which is also the smallest Brazilian number (A125134) and the smallest Brazilian prime (A085104), and as such belongs to A329383.
a(2) = 8 is the smallest composite Brazilian number and so the smallest even composite Brazilian with 8 = 22_3 (A220571).
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.
121 is the only square of prime which is Brazilian with 121 = 11111_3.
In this sequence, there are:
1) The Brazilian primes (except for 31 and 8191) and the only square of prime 121 which are all repunits in a base >= 2 with a string of at least three 1's.
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.
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REFERENCES
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D. Lignon, Dictionnaire de (presque) tous les nombres entiers, Ellipses, 2012, page 420.
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LINKS
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EXAMPLE
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13 = 111_3; 127 = 1111111_2.
20 = 2 * 10 = 22_9; 55 = 5 * 11 = 55_10; 69 = 3 * 23 = 33_22.
31 = 11111_2 = 111_5 so 31 is not a term.
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MATHEMATICA
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Select[Range@ 145, Function[n, Count[Range[2, n - 2], b_ /; SameQ @@ IntegerDigits[n, b]] == 1]] (* Michael De Vlieger, Jun 16 2017 *)
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CROSSREFS
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KEYWORD
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nonn,base,easy
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AUTHOR
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STATUS
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approved
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