A125134 -id:A125134 - cp's OEIS frontend

A307745 Perfect powers y^m with y > 1 and m > 1 which are Brazilian repdigits with three or more digits > 1 in some base.

1521, 1600, 2401, 2744, 6084, 17689, 61009, 244036, 294849, 1179396, 1483524, 2653641, 2725801, 2989441, 4717584, 5239521, 7371225, 9591409, 10614564, 11957764, 14447601, 17397241, 18870336, 20277009, 20958084, 23882769, 26904969, 29484900, 38365636, 38825361

Offset: 1

Bernard Schott, Apr 26 2019

The terms of this sequence are solutions y^m of the Diophantine equation a * (b^q - 1)/(b-1) = y^m with 1 < a < b, y >= 2, q >= 3, m >= 2. This equation has been studied by Kustaa A. Inkeri in Acta Arithmetica; some terms of this sequence come from his article where the author limits the study of this equation to bases b <= 100.
The case a = 1 is clarified in A208242; it corresponds to the Nagell-Ljunggren equation.
The sequence has infinitely many terms because the Diophantine equation 3*(x^2+x+1) = y^2 has infinitely many solutions. - Giovanni Resta, Apr 26 2019
The corresponding solutions (x, y) of this Diophantine equation are (A028231, A341671).
The integers y such that y^2 (m = 2) satisfies this equation are in A158235, except 11 and 20 corresponding to a = 1. - Bernard Schott, Apr 27 2019

			3 * (22^3-1)/(22-1) = 39^2 and (333)_22 = 39^2 = 1521.
58 * (99^4-1)/(99-1) = 7540^2 and (AAAA)_99 = 7540^2 = 56851600 where A is the symbol for 58 in base 99.

Kustaa A. Inkeri, On the Diophantine equation a(x^n-1)/(x-1) = y^m, Acta Arithmetica, XXI, 1972.
Index entries for sequences related to Brazilian numbers.

Subsequence of A001597 and of A125134.
Cf. A158235, A208242 (a=1, that is, with repunits).
Cf. A028231, A341671.

rupQ[n_, mx_] := Block[{t, x, p}, p = x^2 + x + 1; While[(t = p /. x -> mx) <= n && Reduce[p == n && x >= mx, x, Integers] === False, p = x*p + 1]; t <= n]; repdQ[n_] := AnyTrue[ Rest@ Most@ Divisors@ n, rupQ[n/#, #+1] &]; ex = 2; up = 10^7; L = {}; While[2^ex <= up, L = Union[L, Parallelize@ Select[ Range[2, Floor[ up^(1/ex)] ]^ex, repdQ]]; ex = NextPrime@ ex]; L (* Giovanni Resta, Apr 27 2019 *)

isokb(n) = for(b=2, n-2, d=digits(n, b); if((#d > 2) && (vecmin(d)==vecmax(d)) && (d[1] > 1), return (1))); 0;
isok(n) = ispower(n) && isokb(n); \\ Michel Marcus, Apr 28 2019

More terms from Giovanni Resta, Apr 26 2019

A325322 Palindromes in base 10 that are Brazilian.

Original entry on oeis.org

7, 8, 22, 33, 44, 55, 66, 77, 88, 99, 111, 121, 141, 161, 171, 202, 212, 222, 232, 242, 252, 262, 272, 282, 292, 303, 323, 333, 343, 363, 393, 404, 414, 424, 434, 444, 454, 464, 474, 484, 494, 505, 515, 525, 535, 545, 555, 565, 575, 585, 595, 606, 616, 626, 636, 646, 656, 666, 676, 686, 696, 707, 717, 737

Offset: 1

Bernard Schott, Apr 20 2019

Among the terms of this sequence, there are (not exhaustive):
- the even palindromes >= 8,
- the palindromes >= 55 that end with 5,
- the palindromes >= 22 with an even number of digits for they are divisible by 11, and also,
- the palindromes that are Brazilian primes such as 7, 757, 30103, ...

			141 = (33)_46 is a palindrome that is Brazilian.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Intersection of A002113 and A125134.
Complement of A325323 with respect to A002113.
Cf. A288068 (subsequence).

brazQ[n_] := Module[{b = 2, found = False}, While[b < n - 1 && Length @ Union[IntegerDigits[n, b]] > 1, b++]; b < n - 1]; Select[Range[1000], 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

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

A336143 Integers that are Brazilian and not Colombian.

Original entry on oeis.org

8, 10, 12, 13, 14, 15, 16, 18, 21, 22, 24, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 43, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 65, 66, 68, 69, 70, 72, 73, 74, 76, 77, 78, 80, 81, 82, 84, 85, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100

Offset: 1

Bernard Schott, Jul 10 2020

There are no squares of primes in the data (all squares of primes are not Brazilian except for 121 that is Brazilian, but 121 is Colombian).

			15 is a term because 15 = 12 + (sum of digits of 12), so 15 is not Colombian and 15 = 33_4, so 15 is Brazilian.

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

Intersection of A125134 (Brazilian) and A176995 (not Colombian).

Cf. A003052 (Colombian), A333858 (Brazilian and Colombian), this sequence (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 = 100; Select[Union@Table[Plus @@ IntegerDigits[k] + k, {k, 1, n}], # <= n && brazQ[#] &] (* Amiram Eldar, Jul 10 2020 *)

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.

A341058 Numbers that have only one divisor that is Brazilian.

Original entry on oeis.org

7, 8, 10, 12, 13, 15, 18, 22, 27, 31, 33, 34, 38, 43, 46, 49, 51, 55, 57, 58, 69, 73, 74, 82, 85, 87, 94, 95, 106, 111, 115, 118, 121, 122, 123, 125, 127, 134, 141, 142, 145, 157, 158, 159, 166, 169, 177, 178, 183, 185, 187, 194, 201, 202, 205, 206, 209, 211, 213, 214, 218

Offset: 1

Bernard Schott, Feb 15 2021

m is a term iff m is a Brazilian prime (A085104), or m is the square of a Brazilian prime, or m = 121, the only square of prime that is Brazilian, or m = p*q >= 10 with p>q are non-Brazilian primes, or m is the cube of a Brazilian prime, or m = 12 or 18 (see corresponding examples).

			One example for each type of terms that has k divisors:
-> k=2: 7 is a Brazilian prime, hence 7 = 111_2  is a term.
-> k=3: 169 has three divisors {1, 13, 169} and 13 = 111_3 is the only divisor of 169 that is Brazilian, hence 169 is a term.
-> k=3: 121 has three divisors {1, 11, 121} and 121 = 11111_3, hence, 121 that is the only square of prime that is Brazilian, is a term.
-> k=4: 34 has four divisors {1, 2, 17, 34} and 34 = 22_16  is the only divisor of 34 that is Brazilian, hence 34 is a term.
-> k=4: 27 has four divisors {1, 3, 9, 27} and 27 = 33_8 is the only divisor of 27 that is Brazilian, hence 27 is a term.
-> k=6: only two cases: 12 and 18, these integers have each 6 divisors and only 12 = 22_5 and 18 = 33_5 are Brazilian.

Cf. A125134, A220627, A308851, A340795, A341057.

Subsequence: A085104.

brazQ[n_] := Module[{b = 2, found = False}, While[b < n - 1 && Length @ Union[IntegerDigits[n, b]] > 1, b++]; b < n - 1]; Select[Range[200], DivisorSum[#, 1 &, brazQ[#1] &] == 1 &] (* Amiram Eldar, Feb 16 2021 *)

A340795(a(n)) = 1.

A288068 Repdigits in base 10 which are Brazilian numbers.

Original entry on oeis.org

7, 8, 22, 33, 44, 55, 66, 77, 88, 99, 111, 222, 333, 444, 555, 666, 777, 888, 999, 1111, 2222, 3333, 4444, 5555, 6666, 7777, 8888, 9999, 11111, 22222, 33333, 44444, 55555, 66666, 77777, 88888, 99999, 111111, 222222, 333333, 444444, 555555, 666666, 777777, 888888, 999999

Offset: 1

Bernard Schott, Jun 05 2017

These numbers are all repdigits belonging to A010785. The representation of the numbers 7 and 8 in base 10 is not Brazilian but they are yet Brazilian because 7 = 111_2 and 8 = 22_3. Except the repdigits 7, 8, 22, 33, 55, 77 and the primes repunits R_n from A004022 and A004023, all these Brazilian repdigits are also Brazilian in another base.

Contains all base-10 repdigits (A010785) >= 22, because these are Brazilian numbers in base 10. - R. J. Mathar, Jul 19 2024

			7 = 111_2;
44 = 44_10 = 22_21;
66 = 66_10 = 33_21 = 22_32.

Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 0, 11, 0, 0, 0, 0, 0, 0, 0, 0, -10).

Cf. A004022, A004023, A010785, A014181, A125134,

# reuses code of A125134, b-file output
n := 1 :
for ndigs from 1 do
    for d from 1 to 9 do
        r := add(d*10^i,i=0..ndigs-1) ; # rep digit d in base 10
        if isA125134(r) then
            printf("%d %d\n",n,r) ;
            n := n+1 ;
        end if;
    end do:
end do: # R. J. Mathar, Jul 19 2024

Select[Flatten@ Table[FromDigits@ ConstantArray[k, n], {n, 6}, {k, 9}], Function[n, Length@ SelectFirst[Range[2, n - 2], Count[DigitCount[n, #], ?(# > 0 &)] == 1 &] == 0]] (* _Michael De Vlieger, Jun 06 2017, Version 10 *)

A290969 The least positive integer that is a repdigit with length > 2 in exactly n bases.

Original entry on oeis.org

1, 7, 31, 32767, 4095, 435356467, 16777215, 68719476735, 281474976710655

Offset: 0

Bernard Schott, Aug 16 2017

a(10) <= 1152921504606846975 = 2^60 - 1.
a(7) and following terms > 3*10^9. - Giovanni Resta, Aug 16 2017
a(7) <= 2^36-1 and a(8) <= 2^48-1. - Michel Marcus, Aug 17 2017
In fact, we have equality in both cases. - Rémy Sigrist, Aug 21 2017
Except for a(5) = (6^12 - 1) / 5, all the numbers in the data through a(8) are Mersenne numbers A000225. - Bernard Schott, Aug 27 2017

			a(1) = 7 = 111_2.
a(2) = 31 = 11111_2 = 111_5.
a(3) = 32767 = (R_15)_2 = 77777_8 = (31,31,31)_32.

Cf. A000225, A053696, A125134, A167782, A167783, A290869.

a(7) from Rémy Sigrist, Aug 19 2017

a(8) from Rémy Sigrist, Aug 21 2017

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

A307745 Perfect powers y^m with y > 1 and m > 1 which are Brazilian repdigits with three or more digits > 1 in some base.

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A325322 Palindromes in base 10 that are Brazilian.

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A325323 Palindromes in base 10 that are not Brazilian.

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A336143 Integers that are Brazilian and not Colombian.

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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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A341058 Numbers that have only one divisor that is Brazilian.

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A288068 Repdigits in base 10 which are Brazilian numbers.

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