A190300 -id:A190300 - 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

A220570 Numbers that are not Brazilian numbers.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 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

Offset: 1

Bernard Schott, Dec 16 2012

From Bernard Schott, Apr 23 2019: (Start)
The terms of this sequence are:
- integer 1
- oblong semiprime 6,
- primes that are not Brazilian, they are in A220627, and,
- squares of all the primes, except 121 = (11111)_3.
So there is an infinity of integers that are not Brazilian numbers. (End)
This sequence has density 0 as A125134(n) ~ n where A125134 is the complement of this sequence. - David A. Corneth, Jan 22 2021

			25 is a member because it's not possible to write 25=(mm...mm)_b where b is a natural number with 1 < b < 24 and 1 <= m < b.

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

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Bernard Schott, Les nombres brésiliens, Quadrature, no. 76, avril-juin 2010, &6 page 36; included here with permission from the editors of Quadrature.

Cf. A125134 (Brazilian numbers), A190300 (composite numbers not Brazilian), A258165 (odd numbers not Brazilian), A220627 (prime numbers not Brazilian).

for(n=1,300,c=0;for(b=2,n-2,d=digits(n,b);if(vecmin(d)==vecmax(d),c=n;break);c++);if(c==max(n-3,0),print1(n,", "))) \\ Derek Orr, Apr 30 2015

A220571 Composite numbers that are Brazilian.

Original entry on oeis.org

8, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100

Offset: 1

Bernard Schott, Dec 16 2012

nonn

There are just two differences of members with A080257:
1) the term 6 is missing here because 6 is not a Brazilian number.
2) the new term 121 is present although 121 has only 3 divisors, because 121 = 11^2 = 11111_3 is a composite number which is Brazilian. 121 is the lone square of a prime which is Brazilian: Theorem 5, page 37 of Quadrature article in links.
There is an infinity of Brazilian composite numbers (Theorem 1, page 32 of Quadrature article in links: every even number >= 8 is a Brazilian number).

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Bernard Schott, Les nombres brésiliens, Quadrature, no. 76, avril-juin 2010, pages 30-38; included here with permission from the editors of Quadrature.

Cf. A125134, A190300.

Select[Range[4, 10^2], And[CompositeQ@ #, Module[{b = 2, n = #}, While[And[b < n - 1, Length@ Union@ IntegerDigits[n, b] > 1], b++]; b < n - 1]] &] (* Michael De Vlieger, Jul 30 2017, after T. D. Noe at A125134 *)

A340796 a(n) is the smallest number with exactly n divisors that are Brazilian.

Original entry on oeis.org

1, 7, 14, 24, 40, 48, 60, 84, 140, 144, 120, 168, 252, 700, 240, 336, 560, 360, 420, 672, 1120, 2304, 960, 720, 1008, 1080, 840, 2184, 1800, 1260, 2016, 5376, 8960, 2160, 1680, 2880, 4032, 3600, 7056, 19600, 3960, 2520, 3360, 6480, 9072, 9900, 6300, 11520, 16128

Offset: 0

Bernard Schott, Jan 21 2021

Primes can be partitioned into Brazilian primes and non-Brazilian primes. If two distinct primes each larger than 11 are in the same category then the larger one has a multiplicity that is smaller than or equal to that of the smaller prime. - David A. Corneth, Jan 24 2021

			Of the eight divisors of 24, three are Brazilian numbers: 8, 12 and 24, and there is no smaller number with three Brazilian divisors, hence a(3) = 24.

David A. Corneth, Table of n, a(n) for n = 0..427
David A. Corneth, More terms
Wikipédia, Nombre brésilien (in French).
Index entries for sequences related to Brazilian numbers.

Cf. A085104, A125134, A190300, A220570, A308851, A340795, A340797.

Similar with: A087997 (palindromes), A333456 (Niven), A335038 (Zuckerman).

brazQ[n_] := Module[{b = 2, found = False}, While[b < n - 1 && Length[Union[IntegerDigits[n, b]]] > 1, b++]; b < n - 1]; d[n_] := DivisorSum[n, 1 &, brazQ[#] &]; m = 30; s = Table[0, {m}]; c = 0; n = 1; While[c < m, i = d[n]; If[i < m && s[[i + 1]] == 0, c++; s[[i + 1]] = n]; n++]; s (* Amiram Eldar, Jan 21 2021 *)

isokb(n) = for(b=2, n-2, d=digits(n, b); if(vecmin(d)==vecmax(d), return(1))); \\ A125134
isok(k, n) = sumdiv(k, d, isokb(d)) == n;
a(n) = my(k=1); while (!isok(k, n), k++); k; \\ Michel Marcus, Jan 23 2021

More terms from Amiram Eldar, Jan 21 2021

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

A326708 Non-Brazilian squares of primes.

Original entry on oeis.org

4, 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

Offset: 1

Bernard Schott, Aug 26 2019

This sequence is a subsequence of A326707.
For these terms, we have the relations beta'(p^2) = beta"(p^2) = beta(p^2) = (tau(p^2) - 3)/2 = 0.
This sequence = A001248 \ {121} because 121 is the only known square of a prime that is Brazilian (Wikipédia link); 121 is a solution y^q of the Nagell-Ljunggren equation y^q = (b^m-1)/(b-1) with y = 11, q =2, b = 3, m = 5 (see A208242), hence 121 = 11^2 = (3^5 -1)/2 = 11111_3.
The corresponding square roots are: 2, 3, 5, 7, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, ...

			49 = 7^2 is not Brazilian, so beta(49) = 0 with tau(49) = 3.

Bernard Schott, Array relations beta = f(tau) for squares
Wikipédia, 121 (nombre) (in French)
Wikipédia, Nombre brésilien (in French)
Index entries for sequences related to Brazilian numbers

Cf. A190300.
Subsequence of A000290 and of A220570 and of A190300.
Intersection of A001248 and A326707.

brazBases[n_] := Select[Range[2, n - 2], Length[Union[IntegerDigits[n, #]]] == 1 &]; Select[Range[2, 1000], PrimeQ[#^(1/2)]&& brazBases[#] == {} &] (* Metin Sariyar, Sep 05 2019 *)

A364421 For n >= 3, r >= 0, y an integer, a(n) is the number of integral solutions to the elliptic equation y^2 = n^3 + n^2 + 2rn + r^2.

Original entry on oeis.org

2, 2, 2, 3, 2, 4, 3, 4, 2, 8, 2, 4, 8, 5, 2, 7, 2, 9, 8, 4, 2, 15, 3, 4, 5, 10, 2, 15, 2, 7, 8, 4, 8, 17, 2, 4, 8, 15, 2, 15, 2, 10, 14, 4, 2, 22, 3, 7, 8, 9, 2, 10, 8, 15, 8, 4, 2, 38, 2, 4, 14, 8, 8, 15, 2, 9, 7, 16, 2, 27, 2, 4, 13, 9, 8, 15, 2, 22, 6, 4, 2, 39, 8, 4, 7, 16, 2, 27, 8, 10

Offset: 3

Ctibor O. Zizka, Sep 01 2023

nonn

The equation y^2 = n^3 + A*n^2 + B*n + C, where A = 1, B = 2*r, C = r^2 is a minimal model of an elliptic curve with integral coefficients, for details see the Links section.
For a prime number p >= 5, the equation y^2 = p^3 + (p + r)^2 has 2 solutions, r_1 = p*(p - 3)/2 and r_2 = (p + 1)*(p^2 - p - 1)/2.
Factoring the equation y^2 = n^3 + n^2 + 2*r*n + r^2 yields (y+n+r)*(y-n-r) = n^3, which implies y+n+r = d and y-n-r = n^3/d for some divisor d of n^3.  Thus a(n) is the number of divisors d of n^3 such that (d-n^3/d)/2 - n is a nonnegative integer.  This resolves some of Thomas Scheuerle's conjectures. - Robin Visser, Sep 30 2023

			n = 6: y^2 = 6^3 + (6 + r)^2 is valid for r = 9, 19, 47, thus a(6) = 3. The 3 solutions [y, n, n+r] are [21, 6, 15], [29, 6, 25], [55, 6, 53].

Robin Visser, Table of n, a(n) for n = 3..10000
Joseph H. Silverman, The Arithmetic of Elliptic Curves

Cf. A081119, A134108, A190300, A196226, A245685, A364248.

a(n) = length(select((x) -> x[1] >= 0 && x[2] >= n, thue(thueinit(x^2-1,1),n^3),1)) \\ Thomas Scheuerle, Sep 03 2023

def a(n):
    num_sols = 0
    for d in Integer(n^3).divisors():
        if ((d-n^3/d)%2 == 0) and ((d-n^3/d)/2 >= n): num_sols += 1
    return num_sols  # Robin Visser, Sep 30 2023

a(p) = 2 for p prime >= 5, see Comments.
From Thomas Scheuerle, Sep 04 2023: (Start)
Conjecture: a(A190300(n)) = 3.
Conjecture: a(A196226(n)) = 4.
Conjecture: a(p^3) = 5 if p is an odd prime.
Conjecture: a(2*p^2) = 7 if p is an odd prime. But there exist other cases too, for example a(3*23) = 7.
Conjecture: a(prime(n)^prime(n)) = A245685(n - 1) - 1. (End)

a(61)-a(92) from Thomas Scheuerle, Sep 01 2023

A307507 Brazilian semiprimes.

Original entry on oeis.org

10, 14, 15, 21, 22, 26, 33, 34, 35, 38, 39, 46, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, 95, 106, 111, 115, 118, 119, 121, 122, 123, 129, 133, 134, 141, 142, 143, 145, 146, 155, 158, 159, 161, 166, 177, 178, 183, 185, 187, 194, 201, 202, 203, 205, 206, 209, 213, 214, 215, 217, 218

Offset: 1

Bernard Schott, Apr 11 2019

Comparison with A001358 (semiprimes): in this sequence, there are no squared primes apart from 121 = (11111)_3, and also 6 is missing from here since it is not Brazilian.

Different from the squarefree semiprimes of A006881: this sequence = {A006881 \ 6} Union {121}.

			a(20) = 74 = 2 * 37 = (22)_36 is semiprime and Brazilian.
25 = 5 * 5 is semiprime and no Brazilian, and 45 = (55)_8 = (33)_14 = 3^2 * 5 is Brazilian but no semiprime.

Intersection of A001358 and A125134.

Cf. A006881, A190300, A196104.

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.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

Extensions

A220570 Numbers that are not Brazilian numbers.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

PARI

A220571 Composite numbers that are Brazilian.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A340796 a(n) is the smallest number with exactly n divisors that are Brazilian.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A258165 Odd non-Brazilian numbers > 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

A326708 Non-Brazilian squares of primes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A364421 For n >= 3, r >= 0, y an integer, a(n) is the number of integral solutions to the elliptic equation y^2 = n^3 + n^2 + 2*r*n + r^2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

A364421 For n >= 3, r >= 0, y an integer, a(n) is the number of integral solutions to the elliptic equation y^2 = n^3 + n^2 + 2rn + r^2.