A257521 -id:A257521 - cp's OEIS frontend

A125134 "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.

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

Offset: 1

Bernard Schott, Jan 21 2007

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

			15 is a member since it is 33 in base 4.

Pierre Bornsztein, "Hypermath", Vuibert, Exercise a35, p. 7.

Nathaniel Johnston, Table of n, a(n) for n = 1..4000
9th Iberoamerican Mathematical Olympiad, Problem 1: sensible numbers, Fortaleza, Brazil, September 17-25, 1994.
Bernard Schott and others, Postings to the French mathematical forum les-mathematiques.net
Bernard Schott, Les nombres brésiliens, Quadrature, no. 76, avril-juin 2010, pages 30-38; included here with permission from the editors of Quadrature.

Cf. A190300 and A257521 (odd Brazilian numbers).

Cf. A085104 (prime Brazilian numbers).

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

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 *)

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

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

a(n) ~ n. - Charles R Greathouse IV, Aug 09 2017

More terms from Nathaniel Johnston, May 24 2011

A290017 Brazilian numbers which have exactly four Brazilian representations.

Original entry on oeis.org

40, 48, 63, 72, 90, 112, 114, 132, 162, 170, 176, 208, 222, 266, 285, 304, 306, 366, 368, 380, 399, 405, 438, 455, 464, 496, 512, 518, 555, 567, 592, 650, 651, 656, 665, 682, 686, 688, 752, 762, 812, 848, 891, 915, 931, 942, 944, 976, 992, 999, 1024, 1029, 1053, 1072, 1106, 1136, 1168

Offset: 1

Bernard Schott, Jul 28 2017

These numbers could be called 4-Brazilian numbers.
All these numbers are composite with six to twelve divisors.
The smallest number of this sequence is 40 with 40 = 1111_3 = 55_7 = 44_9 = 22_19. The number 40 is a highly Brazilian number in A329383.

			48 = 6 * 8 = 66_7 = 4 * 12 = 44_11 = 3 * 16 = 33_15 = 2 * 24 = 22_23.
63 = 111111_2 = 3 * 21 = 33_20 = 333_4 = 7 * 9 = 77_8.

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

Cf. A125134, A220570, A220571, A257521, A288783, A290015, A290016, A290018, A329383.

Flatten@ Position[#, 4] &@ Table[Count[Range[2, n - 2], ?(And[Length@ # != 1, Length@ Union@ # == 1] &@ IntegerDigits[n, #] &)], {n, 10^3}] (* _Michael De Vlieger, Jul 30 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++>4, return(0))); ct==4 \\ Charles R Greathouse IV, Aug 10 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 *)

A290016 Brazilian numbers which have exactly three Brazilian representations.

Original entry on oeis.org

24, 36, 42, 54, 66, 70, 78, 88, 100, 102, 104, 105, 124, 128, 130, 135, 136, 138, 152, 154, 165, 171, 172, 174, 182, 184, 186, 189, 190, 195, 196, 225, 230, 231, 232, 238, 242, 246, 248, 250, 256, 258, 272, 282, 286, 290, 292, 296, 297, 310, 318, 322, 328, 333, 344, 345

Offset: 1

Bernard Schott, Jul 27 2017

nonn

These numbers could be called 3-Brazilian numbers.
All these numbers are composite with six to ten different divisors.
The smallest number of this sequence is 24 with 24 = 44_5 = 33_7 = 22_11. The number 24 is highly Brazilian in A329383.

			36 = 4 * 9 = 44_8 = 3 * 12 = 33_11 = 2 * 18 = 22_19.
42 = 2 * 21 = 22_20 = 222_4 = 3 * 14 = 33_13.
124 = 4 * 31 = 44_30 = 444_5 = 2 * 62 = 22_61.
272 = 8 * 34 = 88_33 = 4 * 68 = 44_67 = 2 * 136 = 22_135.

Cf. A125134, A220570, A220571, A257521, A288783, A290015, A290017, A290018, A329383.

Flatten@ Position[#, 3] &@ Table[Count[Range[2, n - 2], ?(And[Length@ # != 1, Length@ Union@ # == 1] &@ IntegerDigits[n, #] &)], {n, 350}] (* _Michael De Vlieger, Jul 30 2017 *)

A253261 Odd Brazilian squares.

Original entry on oeis.org

81, 121, 225, 441, 625, 729, 1089, 1225, 1521, 2025, 2401, 2601, 3025, 3249, 3969, 4225, 4761, 5625, 5929, 6561, 7225, 7569, 8281, 8649, 9025, 9801, 11025, 12321, 13225, 13689, 14161, 14641, 15129, 15625, 16641, 17689, 18225, 19881, 20449, 21025, 21609, 23409, 24025, 25281, 25921, 27225, 28561

Offset: 1

Derek Orr, Apr 30 2015

121 is believed to be the only number of the form p^2 for prime p.

The previous comment conjectures the 1 and the 121 are the only difference with respect to A062532. - R. J. Mathar, Jul 25 2015

Cf. A125134, A257521.

for(n=4, 10^5, for(b=2, n-2, d=digits(n, b); if(vecmin(d)==vecmax(d)&&(n+1)%2==0&&issquare(n), print1(n, ", "); break)))

A258165 Odd non-Brazilian numbers > 1.

Original entry on oeis.org

3, 5, 9, 11, 17, 19, 23, 25, 29, 37, 41, 47, 49, 53, 59, 61, 67, 71, 79, 83, 89, 97, 101, 103, 107, 109, 113, 131, 137, 139, 149, 151, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 223, 227, 229, 233, 239, 251, 257, 263, 269, 271, 277, 281, 283, 289, 293, 311, 313, 317, 331, 337

Offset: 1

Daniel Lignon and Robert G. Wilson v, May 22 2015

nonn

Complement of A257521 in A144396 (odd numbers > 1).
The terms are only odd primes or squares of odd primes.
Most odd primes are present except those in A085104.
All terms which are not primes are squares of odd primes except 121 = 11^2.

			11 is present because there is no base b < 11 - 1 = 10 such that the representation of 11 in base b is a repdigit (all digits are equal). In fact, we have: 11 = 1011_2 = 102_3 = 23_4 = 21_5 = 15_6 = 14_7 = 13_8 = 12_9, and none of these representations are repdigits. - _Bernard Schott_, Jun 21 2017

Vincenzo Librandi, Table of n, a(n) for n = 1..1150

Cf. A085104, A125134, A190300, A220570, A220571, A220627, A242397, A253260, A253261, A257521.

fQ[n_] := Block[{b = 2}, While[b < n - 1 && Length@ Union@ IntegerDigits[n, b] > 1, b++]; b+1 == n]; Select[1 + 2 Range@ 170, fQ]

forstep(n=3, 300, 2, c=1; for(b=2, n-2, d=digits(n, b); if(vecmin(d)==vecmax(d), c=0;  break));if(c,print1(n,", "))) \\ Derek Orr, May 27 2015

from sympy.ntheory.factor_ import digits
l=[]
for n in range(3, 301, 2):
    c=1
    for b in range(2, n - 1):
        d=digits(n, b)[1:]
        if max(d)==min(d):
            c=0
            break
    if c: l.append(n)
print(l) # Indranil Ghosh, Jun 22 2017, after PARI program

cp's OEIS Frontend

A125134 "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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A290017 Brazilian numbers which have exactly four Brazilian representations.

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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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A253261 Odd Brazilian squares.

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A258165 Odd non-Brazilian numbers > 1.

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