A329383 -id:A329383 - cp's OEIS frontend

A309493 Highly Brazilian numbers (A329383) that are not highly composite numbers (A002182).

7, 15, 40, 336, 1440, 5405400

Offset: 1

Bernard Schott, Aug 04 2019

Is this sequence finite or infinite?
Indeed, from 6486480 to 321253732800, that is, during 41 successive terms (maybe more?), highly Brazilian numbers are the same as highly composite numbers.
The data for this sequence comes from the new terms in the b-file of A066044 found by Giovanni Resta.
Why are these six numbers HB (highly Brazilian) and not HC (highly composite)? (See link Why HB and not HC? for more details)
1) For 7, 15 and 40, it is because they have a Brazilian representation with 3 or 4 digits and belong to A326380 (see examples).
2) For 336, 1440 and 5405400, it is because each of these three terms HB r is non-oblong, belong to A326386 and the greatest HC m less than r is oblong with the same number of divisors.
a(7) > A329383(91) = 321253732800.

			a(1) = 7 because 7 is the smallest Brazilian number with 7 = 111_2 so beta(7) = 1, as tau(7) = tau(2) = 2, 7 is highly Brazilian but cannot be highly composite.
a(2) = 15 because 15 is the smallest integer 2-Brazilian with 15 = 1111_2 = 33_4 and beta(15) = 2, as tau(15) = tau(6) = 4, 15 is highly Brazilian but not highly composite.
a(3) = 40 because 40 is the smallest integer 4-Brazilian with 40 = 1111_3 = 55_7 = 44_9 = 22_19 so beta(40) = 4, as tau(40) = tau(24) = 8, 40 is highly Brazilian but not highly composite.
a(4) = 336 because beta(336) = 9 and tau(336) = tau(240) = 20.
a(5) = 1440 because beta(1440) = 17 and tau(1440) = tau(1260) = 36.
a(6) = 5405400 because beta(5405400) = 191 and tau(5405400) = tau(4324320) = 384.

Bernard Schott, Why HB and not HC?

Cf. A002182, A279930, A309039, A329383.

A309039 Highly composite numbers (A002182) that are not highly Brazilian (A329383).

Original entry on oeis.org

2, 4, 6, 12, 36, 48, 240

Offset: 1

J. Lowell, Jul 08 2019

Is there a proof that this sequence is infinite?

Indeed, from 1680 to 2882880, that is, during 26 successive terms (maybe more?), highly composite numbers are the same as highly Brazilian numbers. - Bernard Schott, Jul 12 2019

			2 is a highly composite number (A002182) but is not in A329383 (where 1 is followed immediately by 7), so 2 is a term of this sequence.
48 is highly composite with tau(48) = 10, and 48 = 66_7 = 44_11 = 33_15 = 22_33 so beta(48) = 4. We have also beta(40) = 4 with 40 = 1111_3 = 55_7 = 44_9 = 22_19 so 48 is not highly Brazilian. 48 is a term because it is highly composite but not highly Brazilian. - _Bernard Schott_, Jul 12 2019

Cf. A002182 (highly composites), A329383 (highly Brazilian numbers), A279930 (highly composites and highly Brazilian numbers), A309493 (highly Brazilian numbers not highly composites).

A371812 Number of different ways A329383(n) is Brazilian.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 7, 8, 9, 11, 14, 15, 16, 17, 19, 23, 29, 31, 35, 39, 41, 44, 47, 49, 52, 59, 63, 71, 79, 83, 89, 95, 99, 107, 111, 119, 127, 143, 159, 167, 179, 190, 191, 199, 215, 223, 239, 251, 255, 287, 299, 319, 335, 359, 383, 399, 431, 447, 479, 503

Offset: 1

Daniel Mondot, Apr 06 2024

Or number of bases from 2 to n-2 in which the highly Brazilian number A329383(n) is a repdigit number.

Note that from a(26) to a(75) except for a(31) and a(42), a(n) = 4k-1 for some k.

			A329383(2) = 7 is Brazilian in only 1 base below 6 (base 2), so a(2) = 1.
A329383(3) = 15 is Brazilian in 2 bases below 14 (bases 2, 4), so a(3) = 2.

Daniel Mondot, Table of n, a(n) for n = 1..91

Cf. A329383, A284758, A242397.

#include 
FILE *fi;
unsigned int cnt, d, line;
long unsigned base, k, ln;
void main()
{
    fi = fopen("b329383.txt", "r");
    for(;;)
    {
        if (fscanf(fi, "%u %lu", &line, &ln) < 2) break;
        for (cnt=0, base=2; base

A284758 The least positive integer that has exactly n different representations as Brazilian number.

Original entry on oeis.org

1, 7, 15, 24, 40, 60, 144, 120, 180, 336, 420, 360, 900, 960, 720, 840, 1260, 1440, 2340, 1680, 2880, 3600, 8190, 2520, 9072, 9900, 6300, 6720, 20592, 5040, 10920, 7560, 31320, 98040, 25920, 10080, 21420, 177156, 74256, 15120, 28560, 20160

Offset: 0

Bernard Schott, Apr 04 2017

The representation n = 11_(n-1) is not accepted under the definition of a Brazilian number.

The records of this sequence are the highly Brazilian numbers; hence, this sequence is a supersequence of A329383.

			a(0) = 1 because 1 is the smallest non-Brazilian number.
a(4) = 40 because 40 = 1111_3 = 55_7 = 44_9 = 22_19 and 40 is the smallest integer with four Brazilian representations.

D. Lignon, Dictionnaire de (presque) tous les nombres entiers, Ellipses, 2012, page 420.

Giovanni Resta, Table of n, a(n) for n = 0..100

Cf. A066460, A125134, A329383.

rep[n_] := Length@ Select[Range[2, n/2], 1 == Length@ Union@ IntegerDigits[n, #] &]; a[n_] := Block[{k=1}, While[rep[k] != n, k++]; k]; a /@ Range[0, 15] (* Giovanni Resta, Apr 04 2017 *)

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

Original entry on oeis.org

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

A279930 Numbers which are highly composite and highly Brazilian.

Original entry on oeis.org

1, 24, 60, 120, 180, 360, 720, 840, 1260, 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

Bernard Schott, Apr 12 2017

For a(6) = 360 to a(85) = 321253732800, the last term known today, there are 80 successive highly composite numbers that are also highly Brazilian numbers.
If beta(n) is the number of Brazilian representations of n, as in A284758, we have the following relations:
1) for a(k) = m with k <= 85 except 1, 9, 20 and 47, tau(m) = 2*beta(m) + 2, but,
2) for a(1) = 1, tau(1) = 2*beta(1) + 1, because beta(1) = 0, and,
3) for a(9) = 1260, a(20) = 50400 and a(47) = 4324320, tau(m) = 2*beta(m) + 4 because 1260 = 35*36, 50400 = 224*225 and 4324320 = 2079*2080 are oblong numbers.
These improved comments and the b-file come from the new terms in b-file of A066044 found by Giovanni Resta. - Bernard Schott, Aug 03 2019

			360 is the 13th highly composite number and the 10th highly Brazilian number.
336 is the 9th highly Brazilian number, but is not a highly composite number since tau(336) = tau(240) = 20 and 240 is the 12th highly composite number.
240 is the 12th highly composite number, but is not a highly Brazilian number because beta(240) = beta(180) = 8 and 180 is the 8th highly Brazilian number.

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

Bernard Schott, Table of n, a(n) for n = 1..85

Intersection of A002182 (highly composite) and A329383 (highly Brazilian numbers).

Cf. A284758.

Typo in a(18) corrected by J. Lowell, Jul 08 2019

a(29)-a(35) from Bernard Schott, Jul 12 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

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

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

A309493 Highly Brazilian numbers (A329383) that are not highly composite numbers (A002182).

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A309039 Highly composite numbers (A002182) that are not highly Brazilian (A329383).

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A371812 Number of different ways A329383(n) is Brazilian.

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C

A284758 The least positive integer that has exactly n different representations as Brazilian number.

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

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A279930 Numbers which are highly composite and highly Brazilian.

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A288783 Brazilian numbers which have only one Brazilian representation.

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Mathematica

A290017 Brazilian numbers which have exactly four Brazilian representations.

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