A290017 -id:A290017 - cp's OEIS frontend

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

Bernard Schott, Jun 15 2017

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.

			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.

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

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

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

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

A290018 Numbers with exactly five Brazilian representations: bases 1 < b_1 < b_2 < b_3 < b_4 < b_5 < n-1 such that n is a repdigit in base b_i.

Original entry on oeis.org

60, 80, 84, 96, 108, 126, 140, 150, 156, 160, 198, 200, 204, 220, 224, 234, 255, 260, 273, 276, 294, 308, 315, 340, 342, 348, 350, 352, 372, 392, 414, 416, 460, 476, 486, 490, 492, 495, 500, 516, 522, 525, 544, 550, 558, 564, 572, 580, 608, 620, 636, 644, 675, 693, 708, 726, 735, 736

Offset: 1

Bernard Schott, Aug 07 2017

These numbers could be called 5-Brazilian numbers.
All these numbers are composite with 8 to 13 divisors.
The smallest term is 60 and as such is a highly Brazilian number that belongs to A329383.

			60 = 66_9 = 55_11 = 44_14 = 33_19 = 22_29 and tau(60) = 12.
80 = 2222_3 = 22_39 = 44_19 = 55_15 = 88_9 and tau(80) = 10.
255 = 11111111_2 = 3333_4 = 33_84 = 55_50 = (15 15)_16 and tau(255) = 8.
4096 = (32 32)_127 = (16 16)_255 = 88_511 = 44_1023 = 22_2047 and tau(4096) = 13.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

k-Brazilian numbers: A220570 (0), A288783 (1), A290015 (2), A290016 (3), A290017 (4), this sequence (5).

Cf. A125134, A257521, A329383.

Flatten@ Position[#, 5] &@ Table[Count[Range[2, n - 2], ?(And[Length@ # != 1, Length@ Union@ # == 1] &@ IntegerDigits[n, #] &)], {n, 750}] (* _Michael De Vlieger, Aug 09 2017 *)

is(n)=my(d,ct); for(b=2, n-2, d=digits(n, b); for(i=2, #d, if(d[i]!=d[i-1], next(2))); if(ct++>5, return(0))); ct==5 \\ Charles R Greathouse IV, Aug 09 2017

A290015 Brazilian numbers which have exactly two Brazilian representations.

Original entry on oeis.org

15, 18, 21, 26, 28, 30, 31, 32, 44, 45, 50, 52, 56, 57, 62, 64, 68, 75, 76, 85, 86, 91, 92, 93, 98, 99, 110, 111, 116, 117, 129, 133, 146, 147, 148, 153, 164, 175, 183, 188, 207, 212, 215, 219, 236, 243, 244, 245, 259, 261, 268, 275, 279, 284, 314, 316, 325, 332, 338, 341, 343, 356, 363, 365, 369, 381, 387, 388

Offset: 1

Bernard Schott, Jul 17 2017

These numbers could be called 2-Brazilian numbers.
The smallest number of this sequence is 15 which is also the smallest odd composite Brazilian in A257521 with 15 = 11111_2 = 33_4. The number 15 is highly Brazilian in A329383.
Following the Goormaghtigh conjecture, only two primes, 31 and 8191, which are both Mersenne numbers, are Brazilian in two different bases (A119598).

			18 = 2 * 9 = 22_8 = 3 * 6 = 33_5.
26 = 2 * 13 = 2 * 111_3 = 222_3 = 22_12.
31 = 11111_2 = 111_5;
8191 = 1111111111111_2 = 111_90.

Wikipedia, Goormaghtigh conjecture

Cf. A085104, A125134, A220570, A220571, A257521, A284758, A288783, A290016, A290017, A290018, A329383.

bresilienbaseb:=proc(n,b)
local r,q,coupleq:
if n0 then
return [couple[1]+1,r]
else
return [0,0]
end if
end if
end proc;
bresil:=proc(n)
local b,L,k,t:
k:=0:
for b from 2 to (n-2) do
t:=bresilienbase(n,b):
if t[1]>0 then
k:=k+1
L[k]:=[b,t[1],t[2]]:
end if:
end do:
seq(L[i],i=1..k);
end proc;
nbbresil:=n->nops([bresil(n)]);
#Numbers 2 times Brazilian
for n from 1 to 100 do if nbbresil(n)=2 then print(n,bresil(n)) else fi; od:

Mathematica

Flatten@ Position[#, 2] &@ Table[Count[Range[2, n - 2], ?(And[Length@ # != 1, Length@ Union@ # == 1] &@ IntegerDigits[n, #] &)], {n, 400}] (* _Michael De Vlieger, Jul 18 2017 *)

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A288783 Brazilian numbers which have only one Brazilian representation.

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A290018 Numbers with exactly five Brazilian representations: bases 1 < b_1 < b_2 < b_3 < b_4 < b_5 < n-1 such that n is a repdigit in base b_i.

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A290015 Brazilian numbers which have exactly two Brazilian representations.

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A290016 Brazilian numbers which have exactly three Brazilian representations.

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