A220570 -id:A220570 - cp's OEIS frontend

A340797 Integers whose number of divisors that are Brazilian sets a new record.

1, 7, 14, 24, 40, 48, 60, 84, 120, 168, 240, 336, 360, 420, 672, 720, 840, 1260, 1680, 2520, 3360, 5040, 7560, 10080, 15120, 20160, 25200, 27720, 30240, 43680, 45360, 50400, 55440, 65520, 83160, 98280, 110880, 131040, 166320, 196560, 221760, 262080, 277200, 327600

Offset: 1

Bernard Schott, Jan 24 2021

Corresponding number of Brazilian divisors: 0, 1, 2, 3, 4, 5, 6, 7, 10, 11, 14, 15, 17, 18, 19, 26, ...

Observation: the 58 consecutive highly composite numbers from A002182(12) = 240 to A002182(69) = 2095133040 (maybe more, according to conjectured terms) are also terms of this sequence.

			40 has 8 divisors: {1, 2, 4, 5, 8, 10, 20, 40} of which 4 are Brazilian: {8, 10, 20, 40}. No positive integer smaller than 40 has as many as four Brazilian divisors; hence 40 is a term.

David A. Corneth, Table of n, a(n) for n = 1..85
David A. Corneth, Conjectured terms
Wikipédia, Nombre brésilien (in French).
Index entries for sequences related to Brazilian numbers.

Cf. A002182, A125134, A220570, A308851, A340795, A340796.

Similar with: A093036 (palindromes), A340548 (repdigits), A340549 (repunits), A340637 (Niven), A340638 (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[#] &]; dm = -1; s = {}; Do[d1 = d[n]; If[d1 > dm, dm = d1; AppendTo[s, n]], {n, 1, 1000}]; s (* Amiram Eldar, Jan 24 2021 *)

isb(n) = for(b=2, n-2, d=digits(n, b); if(vecmin(d)==vecmax(d), return(1))); \\ A125134
nbd(n) = sumdiv(n, d, isb(d)); \\ A340795
lista(nn) = {my(m=-1); for (n=1, nn, my(x = nbd(n)); if (x > m, print1(n, ", "); m = x););} \\ Michel Marcus, Jan 24 2021

a(20)-a(36) from Michel Marcus, Jan 24 2021

a(37)-a(44) from Amiram Eldar, Jan 24 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

A325323 Palindromes in base 10 that are not Brazilian.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 9, 11, 101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 787, 797, 919, 929, 10201, 10301, 10501, 10601, 11311, 11411, 12421, 12721, 12821, 13331, 13831, 13931, 14341, 14741, 15451, 15551, 16061, 16361, 16561, 16661, 17471, 17971, 18181, 18481, 19391, 19891, 19991

Offset: 1

Bernard Schott, Apr 20 2019

The terms >= 11 of this sequence are either prime palindromes which are not Brazilian, or square of primes (except 121).

Amiram Eldar, Table of n, a(n) for n = 1..1000

Intersection of A002113 and A220570.
Complement of A325322 with respect to A002113.
Cf. A088882 (Palindromes not repdigits).

brazQ[n_] := Module[{b = 2, found = False}, While[b < n - 1 && Length @ Union[IntegerDigits[n, b]] > 1, b++]; b < n - 1]; Select[Range[20000], PalindromeQ[#] && !brazQ[#] &] (* Amiram Eldar, Apr 14 2021 *)

isb(n) = for(b=2, n-2, my(d=digits(n, b)); if(vecmin(d)==vecmax(d), return(1))); \\ A125134
isp(n) = my(d=digits(n)); d == Vecrev(d); \\ A002113
isok(n) = !isb(n) && isp(n); \\ Michel Marcus, Apr 22 2019

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 *)

A336144 Integers that are Colombian and not Brazilian.

Original entry on oeis.org

1, 3, 5, 9, 53, 97, 233, 277, 367, 389, 457, 479, 547, 569, 613, 659, 727, 839, 883, 929, 1021, 1087, 1109, 1223, 1289, 1447, 1559, 1627, 1693, 1783, 1873, 2099, 2213, 2347, 2437, 2459, 2503, 2549, 2593, 2617, 2683, 2729, 2819, 2953, 3023, 3067, 3089, 3313, 3359

Offset: 1

Bernard Schott, Jul 14 2020

There are no even terms because 2, 4 and 6 are not Colombian as 2 = 1 + (sum of digits of 1), 4 = 2 + (sum of digits of 2) and 6 = 3 + (sum of digits of 3), then every even integer >= 8 is Brazilian.

			233 is a term because 233 is not of the form m + (sum of digits of m) for any m < 233, so 233 is Colombian and there is no Brazilian representation for 233.

Giovanni Resta, Self or Colombian number, Numbers Aplenty.
Wikipédia, Nombre brésilien (in French).
Index to sequences related to Colombian Numbers.
Index to sequences related to Brazilian Numbers.

Intersection of A003052 (Colombian) and A220570 (non-Brazilian).

Cf. A125134 (Brazilian), A333858 (Brazilian and Colombian), A336143 (Brazilian and not Colombian), this sequence (Colombian and not Brazilian).

brazQ[n_] := Module[{b = 2, found = False}, While[b < n - 1 && Length[Union[IntegerDigits[n, b]]] > 1, b++]; b < n - 1]; n = 4000; Select[Complement[Range[n], Union @ Table[Plus @@ IntegerDigits[k] + k, {k, 1, n}]], !brazQ[#] &] (* Amiram Eldar, Jul 14 2020 *)

A341057 Numbers without Brazilian divisors.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 9, 11, 17, 19, 23, 25, 29, 37, 41, 47, 53, 59, 61, 67, 71, 79, 83, 89, 97, 101, 103, 107, 109, 113, 131, 137, 139, 149, 151, 163, 167, 173, 179, 181, 191, 193, 197, 199, 223, 227, 229, 233, 239, 251, 257, 263, 269, 271, 277, 281, 283, 289, 293, 311

Offset: 1

Bernard Schott, Feb 04 2021

The first 16 terms are the first 16 terms of A220570 (non-Brazilian numbers), then a(17) = 53 while A220570(17) = 49.

m is a term iff m = 1, or m = 6, or m is a non-Brazilian prime (A220627) or m is the square of a non-Brazilian prime, except for 121 that is Brazilian (see examples).

			One example for each type of terms that has k divisors:
-> k=1: 1 is the smallest number not Brazilian, hence 1 is the first term.
-> k=2: 17 is a prime non-Brazilian, hence 17 is a term.
-> k=3: 25 has three divisors {1, 5, 25} that are all not Brazilian, hence 25 is another term.
-> k=4: 6 has four divisors {1, 2, 3, 6} that are all not Brazilian, hence 6 is the term that has the largest number of divisors.

Cf. A125134, A340795, A308851, A341058 (with 1 Brazilian divisor).
Subsequence of A220570 (non-Brazilian numbers).
Supersequence of A220627 (non-Brazilian primes).

brazQ[n_] := Module[{b = 2, found = False}, While[b < n - 1 && Length @ Union @ IntegerDigits[n, b] > 1, b++]; b < n - 1]; q[n_] := AllTrue[Divisors[n], ! brazQ[#] &]; Select[Range[300], q] (* Amiram Eldar, Feb 04 2021 *)

isb(n) = for(b=2, n-2, my(d=digits(n, b)); if(vecmin(d)==vecmax(d), return(1))); \\ A125134
isok(n) = fordiv(n, d, if (isb(d), return(0))); return(1); \\ Michel Marcus, Feb 07 2021

A340795(a(n)) = 0.

A326775 For any n >= 0, let b >= 2 be the smallest base where n has all digits equal, say to d; a(n) = d.

Original entry on oeis.org

0, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 4, 1, 2, 3, 4, 1, 3, 1, 4, 3, 2, 5, 4, 1, 2, 3, 1, 1, 2, 1, 4, 5, 2, 1, 6, 1, 5, 3, 4, 1, 6, 5, 4, 1, 2, 1, 6, 1, 2, 1, 4, 5, 6, 1, 4, 3, 7, 1, 6, 1, 2, 5, 4, 7, 6, 1, 2, 3, 2, 1, 7, 1, 2

Offset: 0

Rémy Sigrist, Jul 28 2019

A059711 gives base b.
From Bernard Schott, Aug 17 2019: (Start)
a(n) = 1 iff n is A220570, then n = 11_(n-1) or, n is in A053696, then n = 11..11_b for some base b.
a(n) = 2 if n = 2 * p, p prime >= 5.
a(n) = 3 if n = 3 * p, p prime >= 11.
There are k = 2 equal digits in the representation of n in the corresponding base b, except when n is a term of A167782, in which case the number k of equal digits is >= 3. (End)
n = (b^k - 1)/(b - 1) * a(n) so a(n) | n for n > 0. Furthermore a(n) <= sqrt(n). - David A. Corneth, Aug 21 2019
If b is the smallest base such that n=d*b^k+...+d*b^0 (A059711) (d=a(n) is the repdigit) then n mod b = (d*b^k+...+d*b^0) mod b = (d*b^k+...+d*b^1) mod b + (d*b^0) mod b = 0 + (d*1) mod b. Since d is less than the base we end up with the formula n mod b = d. - Jon Maiga, May 31 2021

			For n = 45:
- we have:
     b  45 in base b  Repdigit ?
     -  ------------  ----------
     2  101101        no
     3  1200          no
     4  231           no
     5  140           no
     6  113           no
     7  63            no
     8  55            yes, with d = 5
- hence a(45) = 5.

David A. Corneth, Table of n, a(n) for n = 0..9999

Cf. A059711, A053696, A060775, A220570.

a(n) = for (b=2, oo, if (#Set(digits(n,b))<=1, return (n%b)))

# with library / without (faster for large n)
from sympy.ntheory import digits
def is_repdigit(n, b): return len(set(digits(n, b)[1:])) == 1
def is_repdigit(n, b):
  if n < b: return True
  n, r = divmod(n, b)
  onlyd = r
  while n > b:
    n, r = divmod(n, b)
    if r != onlyd: return False
  return n == onlyd
def a(n):
  for b in range(2, n+3):
    if is_repdigit(n, b): return n%b
print([a(n) for n in range(87)]) # Michael S. Branicky, May 31 2021

n is a multiple of a(n).

a(n) = n mod A059711(n). - Jon Maiga, May 31 2021

A336307 Numbers that are neither Colombian nor Brazilian.

Original entry on oeis.org

2, 4, 6, 11, 17, 19, 23, 25, 29, 37, 41, 47, 49, 59, 61, 67, 71, 79, 83, 89, 101, 103, 107, 109, 113, 131, 137, 139, 149, 151, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 223, 227, 229, 239, 251, 257, 263, 269, 271, 281, 283, 289, 293, 311, 313, 317, 331

Offset: 1

Bernard Schott, Jul 17 2020

The only even terms are 2, 4 and 6 because 2 = 1 + (sum of digits of 1), 4 = 2 + (sum of digits of 2), 6 = 3 + (sum of digits of 3) so these integers are not Colombian then also, because an even number is Brazilian iff it is >= 8.

A333858, A336143, A336144 and this sequence form a partition of the set of positive integers N* ( A000027).

			For b = 17, there is no repdigit in some base b < 16 equal to 17, hence 17 is not Brazilian and 17 = 13 + (sum of digits of 13) hence 17 is not Colombian, so 17 is a term.

Giovanni Resta, Self or Colombian number, Numbers Aplenty.
Wikipédia, Nombre brésilien.
Wikipedia, Partition of a set.
Index to sequences related to Brazilian Numbers.
Index to sequences related to Colombian Numbers.

Intersection of A220570 (not Brazilian) and A176995 (not Colombian).

Cf. A003052 (Colombian), A125134 (Brazilian), A333858 (Brazilian and Colombian), A336143 (Brazilian not Colombian), A336144 (Colombian not Brazilian).

brazQ[n_] := Module[{b = 2, found = False}, While[b < n - 1 && Length[ Union[ IntegerDigits[n, b]]] > 1, b++]; b < n - 1]; n = 300; Select[Union @ Table[Plus @@ IntegerDigits[k] + k, {k, 1, n}], # <= n && !brazQ[#] &] (* Amiram Eldar, Jul 17 2020 *)

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A340797 Integers whose number of divisors that are Brazilian sets a new record.

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A258165 Odd non-Brazilian numbers > 1.

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A325323 Palindromes in base 10 that are not Brazilian.

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A326708 Non-Brazilian squares of primes.

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A336144 Integers that are Colombian and not Brazilian.

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A341057 Numbers without Brazilian divisors.

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A326775 For any n >= 0, let b >= 2 be the smallest base where n has all digits equal, say to d; a(n) = d.

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A336307 Numbers that are neither Colombian nor Brazilian.

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