A288783 Brazilian numbers which have only one Brazilian representation.

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

Offset: 1

Author

Bernard Schott, Jun 15 2017

Keywords

• nonn
• base
• easy

Comments

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.

Examples

			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.
		

References

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

Links

• Bernard Schott, Les nombres brésiliens, Quadrature, no. 76, avril-juin 2010, pages 30-38.

Crossrefs

Cf. A085104, A125134, A220570, A220571, A284758, A290015, A290016, A290017, A290018, A329383.

Programs

Mathematica

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