A125134 -id:A125134 - cp's OEIS frontend

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A066044 Numbers k that are repdigits in more bases (smaller than k) than any smaller number.

1, 3, 7, 15, 24, 40, 60, 120, 180, 336, 360, 720, 840, 1260, 1440, 1680, 2520, 5040, 7560, 10080, 15120, 20160, 25200, 27720, 45360, 50400, 55440, 83160, 110880, 166320, 221760, 277200, 332640, 498960, 554400, 665280, 720720, 1081080, 1441440, 2162160, 2882880

Offset: 1

Erich Friedman, Dec 29 2001

A repdigit has all digits the same in some base.

The number 3 isn't Brazilian (A125134) because 3 = 11_2 is an expansion of the type n = 11_(n-1), which is forbidden for Brazilian numbers. So, except for 3, all the terms here are highly Brazilian numbers (A329383). - Daniel Lignon, Dec 30 2019

			15 is in the sequence since 15 = 1111_2 = 33_4 = 11_14 and no smaller number is a repdigit in 3 different bases.

D. Lignon, Dictionnaire de (presque) tous les nombres entiers, Editions Ellipses, 2012, see p. 420. [In French.]

Giovanni Resta, Table of n, a(n) for n = 1..92

Cf. A066460, A329383.

a = 0 Range[100]; Do[ c = 1; k = 2; While[ k < n-1, If[ Length[ Union[ IntegerDigits[n, k]]] == 1, c++ ]; k++ ]; If[a[[c]] == 0, a[[c]] = n; Print[c, " = ", n]], {n, 1, 200000} ]

okrepu3(b, target, lim) = {my(k = 3, nb = 0, x); while ((x=(b^k-1)/(b-1)) <= target, if (x==target, nb++); k++); nb;}
dge3(n, d) = {my(nb=0, ndi, limi); for (i=1, #d, ndi = n/d[i]; limi = sqrtint(ndi); for (k=d[i]+1, limi, nb += okrepu3(k, ndi, limi););); nb;}
deq2(n, d) = {my(nb=0, nk); for (k=1, #d\2, nk = (n - d[k])/d[k]; if (nk > d[k], nb++);); nb;}
beta23(n) = {if (n<3, return (0)); my(d=divisors(n)); deq2(n, d) + dge3(n, d);}
lista(nn) = {my(m = -1, nm); for (n=1, nn, if ((nm=beta23(n)) > m, print1(n, ", "); m = nm););} \\ Michel Marcus, Jul 13 2019

More terms from Robert G. Wilson v, Jan 02 2002
Offset changed to 1 by Giovanni Resta, Apr 05 2017
a(1) changed to 1 and new terms a(32)-a(41) from Giovanni Resta, Apr 05 2017

A182253 Nonprime numbers n such that n^2 + n + 1 is prime.

Original entry on oeis.org

1, 6, 8, 12, 14, 15, 20, 21, 24, 27, 33, 38, 50, 54, 57, 62, 66, 69, 75, 77, 78, 80, 90, 99, 105, 110, 111, 117, 119, 138, 141, 143, 147, 150, 153, 155, 161, 162, 164, 168, 176, 188, 189, 192, 194, 203, 206, 209, 215, 218, 231, 236, 245, 246, 266, 272, 278

Offset: 1

Bernard Schott, Dec 18 2012

nonn

All these numbers are in A002384 but not in A053182.
The generated prime numbers n^2 + n + 1 are in A185632.
All the generated numbers n^2 + n + 1 = 111_n are by definition Brazilian numbers: A125134. See Links: "Les nombres brésiliens" - Section V.5 page 35.

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. A002383, A002384, A053182, A053183, A085104, A125134, A185632.

Select[Range@ 280, And[! PrimeQ@ #, PrimeQ[#^2 + # + 1]] &] (* Michael De Vlieger, Jul 30 2017 *)

isok(n) = ! isprime(n) && isprime(n^2 + n + 1); \\ Michel Marcus, Sep 04 2013

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

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

A340795 a(n) is the number of divisors of n that are Brazilian.

Original entry on oeis.org

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

Offset: 1

Bernard Schott, Jan 21 2021

The cases a(n) = 0 and a(n) = 1 are respectively detailed in A341057 and A341058.

			For n = 16, the divisors are 1, 2, 4, 8 and 16. Only 8 = 22_3 and 16 = 22_7 are Brazilian numbers, so a(16) = 2.
For n = 30, the divisors are 1, 2, 3, 5, 6, 10, 15 and 30. Only 10 = 22_4, 15 = 33_4 and 30 = 33_9 are Brazilian numbers, so a(30) = 3.
For n = 49, the divisors are 1, 7 and 49. Only 7 = 111_2 is Brazilian, so a(49) = 1 although 49 that is square of prime <> 121 is not Brazilian.

David A. Corneth, Table of n, a(n) for n = 1..10000
Wikipédia, Nombre brésilien (in French).
Index entries for sequences related to Brazilian numbers.

Cf. A085104, A125134, A220627, A308851, A340796, A340797, A341057, A341058.

brazQ[n_] := Module[{b = 2, found = False}, While[b < n - 1 && Length[Union[IntegerDigits[n, b]]] > 1, b++]; b < n - 1]; a[n_] := DivisorSum[n, 1 &, brazQ[#] &]; Array[a, 100] (* Amiram Eldar, Jan 21 2021 *)

isb(n) = for(b=2, n-2, d=digits(n, b); if(vecmin(d)==vecmax(d), return(1))); \\ A125134
a(n) = sumdiv(n, d, isb(d)); \\ Michel Marcus, Jan 24 2021

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

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Author

Bernard Schott, Jul 27 2017

Keywords

nonn

Comments

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.

Examples

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.

Crossrefs

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

Programs

Mathematica

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 *)

A059711 Smallest base in which n is a repdigit.

Original entry on oeis.org

2, 2, 3, 2, 3, 4, 5, 2, 3, 8, 4, 10, 5, 3, 6, 2, 7, 16, 5, 18, 9, 4, 10, 22, 5, 24, 3, 8, 6, 28, 9, 2, 7, 10, 16, 6, 8, 36, 18, 12, 3, 40, 4, 6, 10, 8, 22, 46, 7, 48, 9, 16, 12, 52, 8, 10, 13, 7, 28, 58, 9, 60, 5, 2, 15, 12, 10, 66, 16, 22, 9, 70, 11, 8, 36, 14, 18, 10, 12, 78, 3, 26, 40, 82, 11, 4

Offset: 0

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Author

Erich Friedman, Feb 19 2001

Keywords

base
easy
nonn

Comments

Numbers n such that a(n) < n - 1 correspond to Brazilian numbers (A125134); conversely, positive numbers n such that a(n) >= n - 1 correspond to A220570. - Rémy Sigrist, Apr 04 2018

Examples

a(13) = 3 since 13 in base 3 is 111.

Links

Rémy Sigrist, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A010785, A125134, A048328, A048329, A048330, A048331, A048332, A048333, A048334, A048335, A048336, A048337, A048338, A048339, A048340, A220570.

Programs

PARI

a(n) = for (b=2, oo, if (#Set(digits(n, b))<=1, return (b))) \\ Rémy Sigrist, Apr 04 2018

Formula

From Rémy Sigrist, Apr 04 2018: (Start)

a(n) <= n - 1 for any n >= 3.

a(2^n-1) = 2 for any n >= 0.

a(A048328(n)) <= 3 for any n >= 0.

a(A048329(n)) <= 4 for any n >= 0.

a(A048330(n)) <= 5 for any n >= 0.

a(A048331(n)) <= 6 for any n >= 0.

a(A048332(n)) <= 7 for any n >= 0.

a(A048333(n)) <= 8 for any n >= 0.

a(A048334(n)) <= 9 for any n >= 0.

a(A010785(n)) <= 10 for any n >= 0.

a(A048335(n)) <= 11 for any n >= 0.

a(A048336(n)) <= 12 for any n >= 0.

a(A048337(n)) <= 13 for any n >= 0.

a(A048338(n)) <= 14 for any n >= 0.

a(A048339(n)) <= 15 for any n >= 0.

a(A048340(n)) <= 16 for any n >= 0.

(End)

Extensions

Example clarified by Harvey P. Dale, Oct 11 2015

Terms a(0) = 2, a(1) = 2 and a(2) = 3 prepended by Rémy Sigrist, Apr 04 2018

A193366 Primes of the form n^4 + n^3 + n^2 + n + 1 where n is nonprime.

Original entry on oeis.org

5, 22621, 245411, 346201, 637421, 837931, 2625641, 3835261, 6377551, 15018571, 16007041, 21700501, 30397351, 35615581, 52822061, 78914411, 97039801, 147753211, 189004141, 195534851, 209102521, 223364311, 279086341, 324842131, 421106401, 445120421, 566124791, 693025471, 727832821, 745720141, 880331261, 943280801, 987082981, 1544755411, 1740422941

Offset: 1

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Author

Jonathan Vos Post, Dec 20 2012

Keywords

nonn
easy

Comments

Note that there are no primes of the form n^3 + n^2 + n + 1 = (n+1)*(n^2+1) as irreducible components over Z.

From Bernard Schott, May 15 2017: (Start)

These are the primes associated with A286094.

A088548 = A190527 Union {This sequence}.

All the numbers of this sequence n^4 + n^3 + n^2 + n + 1 = 11111_n with n > 1 are Brazilian numbers, so belong to A125134 and A085104. (End)

Examples

a(1) = 1^4 + 1^3 + 1^2 + 1 + 1 = 5. a(2) = 12^4 + 12^3 + 12^2 + 12 + 1 = 22621.

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Bernard Schott, Les nombres brésiliens, Reprinted from Quadrature, no. 76, avril-juin 2010, pages 30-38.

Crossrefs

Subsequence of A088548.

Cf. A000040, A018252, A185632, A192321.

Cf. A049409, A053699, A065509, A085104, A088548, A125134, A190527, A286094.

Programs

Maple

for n from 1 to 150 do p(n):= 1+n+n^2+n^3+n^4; if tau(n)>2 and isprime(p(n)) then print(n,p(n)) else fi od: # Bernard Schott, May 15 2017

Mathematica

Select[Map[Total[#^Range[0, 4]] &, Select[Range@ 204, ! PrimeQ@ # &]], PrimeQ] (* Michael De Vlieger, May 15 2017 *)

PARI

print1(5);forcomposite(n=4,1e3,if(isprime(t=n^4+n^3+n^2+n+1),print1(", "t))) \\ Charles R Greathouse IV, Mar 25 2013

Formula

{n^4 + n^3 + n^2 + n + 1 where n is in A018252}.

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A066044 Numbers k that are repdigits in more bases (smaller than k) than any smaller number.

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Extensions

A182253 Nonprime numbers n such that n^2 + n + 1 is prime.

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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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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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A340795 a(n) is the number of divisors of n that are Brazilian.

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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.