A220571 -id:A220571 - cp's OEIS frontend

A000215 Fermat numbers: a(n) = 2^(2^n) + 1.

3, 5, 17, 257, 65537, 4294967297, 18446744073709551617, 340282366920938463463374607431768211457, 115792089237316195423570985008687907853269984665640564039457584007913129639937

Offset: 0

N. J. A. Sloane

It is conjectured that just the first 5 numbers in this sequence are primes.
An infinite coprime sequence defined by recursion. - Michael Somos, Mar 14 2004
For n>0, Fermat numbers F(n) have digital roots 5 or 8 depending on whether n is even or odd (Koshy). - Lekraj Beedassy, Mar 17 2005
This is the special case k=2 of sequences with exact mutual k-residues. In general, a(1)=k+1 and a(n)=min{m | m>a(n-1), mod(m,a(i))=k, i=1,...,n-1}. k=1 gives Sylvester's sequence A000058. - Seppo Mustonen, Sep 04 2005
For n>1 final two digits of a(n) are periodically repeated with period 4: {17, 57, 37, 97}. - Alexander Adamchuk, Apr 07 2007
For 1 < k <= 2^n, a(A007814(k-1)) divides a(n) + 2^k. More generally, for any number k, let r = k mod 2^n and suppose r != 1, then a(A007814(r-1)) divides a(n) + 2^k. - T. D. Noe, Jul 12 2007
From Daniel Forgues, Jun 20 2011: (Start)
The Fermat numbers F_n are F_n(a,b) = a^(2^n) + b^(2^n) with a = 2 and b = 1.
For n >= 2, all factors of F_n = 2^(2^n) + 1 are of the form k*(2^(n+2)) + 1 (k >= 1).
The products of distinct Fermat numbers (in their binary representation, see A080176) give rows of Sierpiński's triangle (A006943). (End)
Let F(n) be a Fermat number. For n > 2, F(n) is prime if and only if 5^((F(n)-1)/4) == sqrt(F(n)-1) (mod F(n)). - Arkadiusz Wesolowski, Jul 16 2011
Conjecture: let the smallest prime factor of Fermat number F(n) be P(F(n)). If F(n) is composite, then P(F(n)) < 3*2^(2^n/2 - n - 2). - Arkadiusz Wesolowski, Aug 10 2012
The Fermat primes are not Brazilian numbers, so they belong to A220627, but the Fermat composites are Brazilian numbers so they belong to A220571. For a proof, see Proposition 3 page 36 on "Les nombres brésiliens" in Links. - Bernard Schott, Dec 29 2012
It appears that this sequence is generated by starting with a(0)=3 and following the rule "Write in binary and read in base 4".  For an example of "Write in binary and read in ternary", see A014118. - John W. Layman, Jul 30 2013
Conjecture: the numbers > 5 in this sequence, i.e., 2^2^k + 1 for k>1, are exactly the numbers n such that (n-1)^4-1 divides 2^(n-1)-1. - M. F. Hasler, Jul 24 2015

			a(0) = 1*2^1 + 1 = 3 = 1*(2*1) + 1.
a(1) = 1*2^2 + 1 = 5 = 1*(2*2) + 1.
a(2) = 1*2^4 + 1 = 17 = 2*(2*4) + 1.
a(3) = 1*2^8 + 1 = 257 = 16*(2*8) + 1.
a(4) = 1*2^16 + 1 = 65537 = 2048*(2*16) + 1.
a(5) = 1*2^32 + 1 = 4294967297 = 641*6700417 = (10*(2*32) + 1)*(104694*(2*32) + 1).
a(6) = 1*2^64 + 1 = 18446744073709551617 = 274177*67280421310721 = (2142*(2*64) + 1)*(525628291490*(2*64) + 1).

M. Aigner and G. M. Ziegler, Proofs from The Book, Springer-Verlag, Berlin, 2nd. ed., 2001; see p. 3.
T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 7.
P. Bachmann, Niedere Zahlentheorie (1902, 1910), reprinted Chelsea, NY, 1968, vol. 2, p. 87.
James Gleick, Faster, Vintage Books, NY, 2000 (see pp. 259-261).
Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §3.2 Prime Numbers, pp. 78-79.
R. K. Guy, Unsolved Problems in Number Theory, A3.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 14.
E. Hille, Analytic Function Theory, Vol. I, Chelsea, N.Y., see p. 64.
T. Koshy, "The Digital Root Of A Fermat Number", Journal of Recreational Mathematics Vol. 32 No. 2 2002-3 Baywood NY.
M. Krizek, F. Luca & L. Somer, 17 Lectures on Fermat Numbers, Springer-Verlag NY 2001.
C. S. Ogilvy and J. T. Anderson, Excursions in Number Theory, Oxford University Press, NY, 1966, p. 36.
Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see pp. 18, 59.
C. A. Pickover, The Math Book, Sterling, NY, 2009; see p. 202.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 6-7, 70-75.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 136-137.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, 1987, pp. 148-149.

N. J. A. Sloane, Table of n, a(n) for n = 0..11
Richard Bellman, A note on relatively prime sequences, Bull. Amer. Math. Soc. 53 (1947), 778-779.
Chris Bispels, Matthew Cohen, Joshua Harrington, Joshua Lowrance, Kaelyn Pontes, Leif Schaumann, and Tony W. H. Wong, A further investigation on covering systems with odd moduli, arXiv:2507.16135 [math.NT], 2025. See p. 3.
Chris K. Caldwell, The Prime Glossary, Fermat number.
Chris K. Caldwell, The prime pages All prime-square Mersenne divisors are Wieferich (2021).
Shane Chern, Fermat Numbers in Multinomial Coefficients, J. Int. Seq. 17 (2014) # 14.3.2.
Leonhard Euler, Observations on a theorem of Fermat and others on looking at prime numbers, arXiv:math/0501118 [math.HO], 2005-2008.
Leonhard Euler, Observationes de theoremate quodam Fermatiano aliisque ad numeros primos spectantibus.
Emmanuel Ferrand, Deformations of the Taylor Formula, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.7.
Richard K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]
Christian Kassel and Christophe Reutenauer, Pairs of intertwined integer sequences, arXiv:2507.15780 [math.NT], 2025. See p. 12.
Wilfrid Keller, Prime factors k.2^n + 1 of Fermat numbers F_m
Jiří Klaška, Jakóbczyk's Hypothesis on Mersenne Numbers and Generalizations of Skula's Theorem, J. Int. Seq., Vol. 26 (2023), Article 23.3.8.
T.-W. Leung, A Brief Introduction to Fermat Numbers
Romeo Meštrović, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv preprint arXiv:1202.3670 [math.HO], 2012. - From _N. J. A. Sloane_, Jun 13 2012
Romeo Meštrović, Goldbach-type conjectures arising from some arithmetic progressions, University of Montenegro, 2018.
Romeo Meštrović, Goldbach's like conjectures arising from arithmetic progressions whose first two terms are primes, arXiv:1901.07882 [math.NT], 2019.
Michael A. Morrison and John Brillhart, The factorization of F_7, Bull. Amer. Math. Soc. 77 (1971), 264.
Robert Munafo, Fermat Numbers
Robert Munafo, Notes on Fermat numbers
Seppo Mustonen, On integer sequences with mutual k-residues.
Seppo Mustonen, On integer sequences with mutual k-residues. [Local copy]
OEIS Wiki, Fermat numbers.
OEIS Wiki, Sierpinski's triangle.
G. A. Paxson, The compositeness of the thirteenth Fermat number, Math. Comp. 15 (76) (1961) 420-420.
Carl Pomerance, A tale of two sieves, Notices Amer. Math. Soc., 43 (1996), 1473-1485.
P. Sanchez, PlanetMath.org, Fermat Numbers
Bernard Schott, Les nombres brésiliens, Quadrature, no. 76, avril-juin 2010, pages 30-38. Local copy, included here with permission from the editors of Quadrature.
G. Villemin's Almanach of Numbers, Nombres de Fermat.
Le Roy J. Warren, Henry G. Bray, On the square-freeness of Fermat and Mersenne Numbers, Pac. J. Math. 22 (3) (1967) 563.
Eric Weisstein's World of Mathematics, Fermat Number.
Eric Weisstein's World of Mathematics, Generalized Fermat Number.
Wikipedia, Fermat number.
Wolfram Research, Fermat numbers are pairwise coprime.

a(n) = A001146(n) + 1 = A051179(n) + 2.
Cf. A019434, A050922, A051158, A051179, A063486, A073617, A085866.
See A004249 for a similar sequence.
Cf. A080176 for binary representation of Fermat numbers.
Cf. A220627, A220570, A220571, A125134, A249119.
Cf. A000058, A000120, A000225, A006943, A007814, A014118, A077585, A242619, A242620, A257916, A257917.
Cf. A000058, A000120, A000225, A006943, A007814, A014118, A077585, A242619, A242620, A257916, A257917.

a000215 = (+ 1) . (2 ^) . (2 ^)  -- Reinhard Zumkeller, Feb 13 2015

Maple
```
A000215 := n->2^(2^n)+1;
```

Table[2^(2^n) + 1, {n, 0, 8}] (* Alonso del Arte, Jun 07 2011 *)

A000215(n):=2^(2^n)+1$ makelist(A000215(n),n,0,10); /* Martin Ettl, Dec 10 2012 */

PARI
```
a(n)=if(n<1,3*(n==0),(a(n-1)-1)^2+1)
```

def a(n): return 2**(2**n) + 1
print([a(n) for n in range(9)]) # Michael S. Branicky, Apr 19 2021

a(0) = 3; a(n) = (a(n-1)-1)^2 + 1, n >= 1.
a(n) = a(n-1)*a(n-2)*...*a(1)*a(0) + 2, n >= 0, where for n = 0, we get the empty product, i.e., 1, plus 2, giving 3 = a(0). - Benoit Cloitre, Sep 15 2002 [edited by Daniel Forgues, Jun 20 2011]
The above formula implies that the Fermat numbers (being all odd) are coprime.
Conjecture: F is a Fermat prime if and only if phi(F-2) = (F-1)/2. - Benoit Cloitre, Sep 15 2002
A000120(a(n)) = 2. - Reinhard Zumkeller, Aug 07 2010
If a(n) is composite, then a(n) = A242619(n)^2 + A242620(n)^2 = A257916(n)^2 - A257917(n)^2. - Arkadiusz Wesolowski, May 13 2015
Sum_{n>=0} 1/a(n) = A051158. - Amiram Eldar, Oct 27 2020
From Amiram Eldar, Jan 28 2021: (Start)
Product_{n>=0} (1 + 1/a(n)) = A249119.
Product_{n>=0} (1 - 1/a(n)) = 1/2. (End)
a(n) = 2*A077585(n) + 3. - César Aguilera, Jul 26 2023
a(n) = 2*2^A000225(n) + 1. - César Aguilera, Jul 11 2024

A023001 a(n) = (8^n - 1)/7.

Original entry on oeis.org

0, 1, 9, 73, 585, 4681, 37449, 299593, 2396745, 19173961, 153391689, 1227133513, 9817068105, 78536544841, 628292358729, 5026338869833, 40210710958665, 321685687669321, 2573485501354569, 20587884010836553, 164703072086692425

Offset: 0

David W. Wilson

Gives the (zero-based) positions of odd terms in A007556 (numbers n such that A007556(a(n)) mod 2 = 1). - Farideh Firoozbakht, Jun 13 2003
{1, 9, 73, 585, 4681, ...} is the binomial transform of A003950. - Philippe Deléham, Jul 22 2005
Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=8, (i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n >= 1, a(n) = det(A). - Milan Janjic, Feb 21 2010
Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=9, (i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n >= 1, a(n-1) = (-1)^n*charpoly(A,1). - Milan Janjic, Feb 21 2010
This is the sequence A(0,1;7,8;2) = A(0,1;8,0;1) of the family of sequences [a,b:c,d:k] considered by G. Detlefs, and treated as A(a,b;c,d;k) in the W. Lang link given below. - Wolfdieter Lang, Oct 18 2010
a(n) is the total number of squares the carpetmaker has removed after the n-th step of a Sierpiński carpet production. - Ivan N. Ianakiev, Oct 22 2013
For n >= 1, a(n) is the total number of holes in a box fractal (start with 8 boxes, 1 hole) after n iterations. See illustration in link. - Kival Ngaokrajang, Jan 27 2015
From Bernard Schott, May 01 2017: (Start)
Except for 0, 1 and 73, all the terms are composite because a(n) = ((2^n - 1) * (4^n + 2^n + 1))/7.
For n >= 3, all terms are Brazilian repunits numbers in base 8, and so belong to A125134.
a(3) = 73 is the only Brazilian prime in base 8, and so it belongs to A085104 and A285017. (End)

			From _Zerinvary Lajos_, Jan 14 2007: (Start)
Octal.............decimal
0....................0
1....................1
11...................9
111.................73
1111...............585
11111.............4681
111111...........37449
1111111.........299593
11111111.......2396745
111111111.....19173961
1111111111...153391689
etc. ...............etc. (End)
a(4) = (8^4 - 1)/7 = 585 = 1111_8 = (2^4 - 1) * (4^4 + 2^4 + 1)/7 = 15 * 273/7 = 15 * 39. - _Bernard Schott_, May 01 2017

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
A. Abdurrahman, CM Method and Expansion of Numbers, arXiv:1909.10889 [math.NT], 2019.
Carlos M. da Fonseca and Anthony G. Shannon, A formal operator involving Fermatian numbers, Notes Num. Theor. Disc. Math. (2024) Vol. 30, No. 3, 491-498.
Roger B. Eggleton, Maximal Midpoint-Free Subsets of Integers, International Journal of Combinatorics Volume 2015, Article ID 216475, 14 pages.
Wolfdieter Lang, Notes on certain inhomogeneous three term recurrences. - _Wolfdieter Lang_, Oct 18 2010
Kival Ngaokrajang, Illustration of initial terms
Quynh Nguyen, Jean Pedersen, and Hien T. Vu, New Integer Sequences Arising From 3-Period Folding Numbers, Vol. 19 (2016), Article 16.3.1. See Table 1.
Kalika Prasad, Munesh Kumari, Rabiranjan Mohanta, and Hrishikesh Mahato, The sequence of higher order Mersenne numbers and associated binomial transforms, arXiv:2307.08073 [math.NT], 2023.
D. C. Santos, E. A. Costa, and P. M. M. C. Catarino, On Gersenne Sequence: A Study of One Family in the Horadam-Type Sequence, Axioms 14, 203, (2025). See p. 4.
Eric Weisstein's World of Mathematics, Repunit.
Index entries for linear recurrences with constant coefficients, signature (9,-8).

Cf. A007556, A003950, A001158, A033138.

Cf. A125134, A085104, A285017, A220571, A053696.

A023001:=List([0..10^2],n->(8^n-1)/7); # Muniru A Asiru, Oct 03 2017

[(8^n-1)/7: n in [0..20]]; // Vincenzo Librandi, Sep 17 2011

a:=n->sum(8^(n-j),j=1..n): seq(a(n), n=0..20); # Zerinvary Lajos, Jan 04 2007

Table[(8^n-1)/7, {n, 0, m}]
LinearRecurrence[{9,-8},{0,1},30] (* Harvey P. Dale, Feb 12 2015 *)

A023001(n):=floor((8^n-1)/7)$
makelist(A023001(n),n,0,30); /* Martin Ettl, Nov 05 2012 */

a(n)=(8^n-1)/7 \\ Charles R Greathouse IV, Mar 22 2016

[lucas_number1(n,9,8) for n in range(0, 21)] # Zerinvary Lajos, Apr 23 2009

[gaussian_binomial(n,1,8) for n in range(0,21)] # Zerinvary Lajos, May 28 2009

Also sum of cubes of divisors of 2^(n-1): a(n) = A001158(A000079(n-1)). - Labos Elemer, Apr 10 2003 and Farideh Firoozbakht, Jun 13 2003
a(n) = A033138(3n-2). - Alexandre Wajnberg, May 31 2005
From Philippe Deléham, Oct 12 2006: (Start)
a(0) = 0, a(n) = 8*a(n-1) + 1 for n>0.
G.f.: x/((1-8x)*(1-x)). (End)
From Wolfdieter Lang, Oct 18 2010: (Start)
a(n) = 7*a(n-1) + 8*a(n-2) + 2, a(0)=0, a(1)=1.
a(n) = 8*a(n-1) + a(n-2) - 8*a(n-3) = 9*a(n-1) - 8*a(n-2), a(0)=0, a(1)=1, a(2)=9. Observation by Gary Detlefs. See the W. Lang comment and link. (End)
a(n) = Sum_{k=0..n-1} 8^k. - Doug Bell, May 26 2017
E.g.f.: exp(x)*(exp(7*x) - 1)/7. - Stefano Spezia, Mar 11 2023

A190300 Composite numbers that are not Brazilian.

Original entry on oeis.org

4, 6, 9, 25, 49, 169, 289, 361, 529, 841, 961, 1369, 1681, 1849, 2209, 2809, 3481, 3721, 4489, 5041, 5329, 6241, 6889, 7921, 9409, 10201, 10609, 11449, 11881, 12769, 16129, 17161, 18769, 19321, 22201, 22801, 24649, 26569, 27889, 29929, 32041, 32761, 36481, 37249, 38809, 39601, 44521, 49729

Offset: 1

N. J. A. Sloane, May 14 2011

nonn

Other than the term 6 and the missing term 121, is this sequence the same as A001248? - Nathaniel Johnston, May 24 2011
From Bernard Schott, Dec 04 2012: (Start)
Yes, because
1) 4 is not a Brazilian number [4 = 100_2].
2) 6 is not a Brazilian number [6 = 110_2 = 20_3 = 12_4].
3) Theorem 1, page 32 of Quadrature article mentioned in links: If n > 7 is not Brazilian, then n is a prime or the square of a prime.
4) Theorem 5, page 37 of Quadrature article mentioned in links: The only square of prime number which is Brazilian is 121 = 11^2 = 11111_3.
(End)
There is an infinity of composite numbers that are not Brazilian: Corollary 2, page 37 of Quadrature article in links (consider the sequence of squares of prime numbers for p >= 13). - Bernard Schott, Dec 17 2012
Also semiprimes that are not Brazilian. - Bernard Schott, Apr 11 2019

			a(10) = p_10^2 = 29^2 = 841.

Robert Israel, Table of n, a(n) for n = 1..10000 (first 208 terms from Robert G. Wilson v)
Bernard Schott, Les nombres brésiliens, Quadrature, no. 76, avril-juin 2010, pages 30-38; included here with permission from the editors of Quadrature.

Cf. A085104, A125134, A189891, A220571, A307507.
Intersection of A002808 and A220570.
Intersection of A001358 and A220570.

4, 6, 9, 25, 49,seq(ithprime(i)^2, i=6..100); # Robert Israel, Apr 17 2019

brazBases[n_] := Select[Range[2, n - 2], Length[Union[IntegerDigits[n, #]]] == 1 &]; Select[Range[2, 10000], ! PrimeQ[#] && brazBases[#] == {} &] (* T. D. Noe, Dec 26 2012 *)
f[n_] := Block[{b = 2}, While[ Length@ Union@ IntegerDigits[n, b] != 1, b++]; b]; k = 4; lst = {}; While[k < 50001, If[ !PrimeQ@ k && 1 + f@ k == k, AppendTo[lst, k]]; k++]; lst (* Robert G. Wilson v, Mar 30 2014 *)

isnotb(n) = my(c=0, d); for(b=2, n-2, d=digits(n, b); if(vecmin(d)==vecmax(d), c=n; break); c++); (c==max(n-3, 0)); \\ A220570
lista(nn) = forcomposite(n=1, nn, if (isnotb(n), print1(n, ", "))); \\ Michel Marcus, Apr 14 2019

a(1) = 2^2 = p_1^2, a(2) = 2*3 = p_1*p_2, a(3) = 3^2 = p_2^2, a(4) = 5^2 = p_3^2, a(5) = 7^2 = p_4^2, a(6) = 13^2 = p_6^2, ..., for n >= 6, a(n) = p_n^2, where p_k is the k-th prime number. - Bernard Schott, Dec 04 2012

a(6)-a(24) from Nathaniel Johnston, May 24 2011

a(25) onward from Robert G. Wilson v, Mar 30 2014

A208242 Perfect powers y^q with y > 1 and q > 1 which are Brazilian repunits with three or more digits in some base.

Original entry on oeis.org

121, 343, 400

Offset: 1

Bernard Schott, Jan 11 2013

These three numbers are the only known solutions y^q of the Nagell-Ljunggren equation (b^m-1)/(b-1) = y^q with y > 1, q > 1, b > 1, m > 2. Yann Bugeaud and Maurice Mignotte propose two alternative conjectures:
A) The Nagell-Ljunggren equation has only these three solutions.
Considering the current state of our knowledge, this conjecture seems too ambitious, while the next one seems more reasonable.
B) The Nagell-Ljunggren equation has only a finite number of solutions.
This last conjecture is true if the abc conjecture is true (see article Bugeaud-Mignotte in link, p. 148).
Consequence: 121 is the only known square of prime which is Brazilian.
There are no other solutions for some base b < 10000.
Some theorems and results about this equation:
With the exception of the 3 known solutions,
1) for q = 2, there are no other solutions than 11^2 and 20^2,
2) there is no other solution if 3 divides m than 7^3,
3) there is no other solution if 4 divides m than 20^2. - Bernard Schott, Apr 29 2019
From David A. Corneth, Apr 29 2019: (Start)
Intersection of A001597 and A053696.
a(4) > 10^25 if it exists using constraints above.
In the Nagell-Ljunggren equation, we need b > 2. If b = 2, we get y^q = 2^m - 1 which by Catalan's conjecture has no solutions (see A001597). (End)

			121 = 11^2 =  (3^5 - 1)/ (3 - 1) = 11111_3.
343 =  7^3 = (18^3 - 1)/(18 - 1) =   111_18.
400 = 20^2 =  (7^4 - 1)/ (7 - 1) =  1111_7.

Y. Bugeaud and M. Mignotte, L'équation de Nagell-Ljunggren (x^n-1)/(x-1) = y^q, Enseign. Math. 48(2002), 147-168.

Cf. A001597, A053696, A220571 (Brazilian composites), A307745 (similar but with digits > 1).

is(n) = if(!ispower(n), return(0)); for(b=2, n-1, my(d=digits(n, b)); if(#d > 2 && vecmin(d)==1 && vecmax(d)==1, return(1))); 0 \\ Felix Fröhlich, Apr 29 2019

Small edits to the name by Bernard Schott, Apr 30 2019

A288783 Brazilian numbers which have only one Brazilian representation.

Original entry on oeis.org

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 *)

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

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 *)

cp's OEIS Frontend

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A208242 Perfect powers y^q with y > 1 and q > 1 which are Brazilian repunits with three or more digits in some base.

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A288783 Brazilian numbers which have only one Brazilian representation.

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A290017 Brazilian numbers which have exactly four Brazilian representations.

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