cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

A220136 Number of ways that a number n can be written as ddd...d where d is a digit in base b with 1 < b < n-1.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 2, 1, 0, 2, 0, 1, 2, 1, 0, 3, 0, 2, 1, 2, 0, 2, 2, 2, 1, 1, 1, 3, 0, 1, 1, 4, 0, 3, 1, 2, 2, 1, 0, 4, 0, 2, 1, 2, 0, 3, 1, 2, 2, 1, 0, 5, 0, 2, 4, 2, 1, 3, 0, 2, 1, 3, 0, 4, 1, 1, 2, 2, 1, 3, 0, 5, 1, 1, 0, 5, 2, 2, 1
Offset: 1

Views

Author

T. D. Noe, Dec 26 2012

Keywords

Comments

When a(n) > 0, n is called Brazilian (A125134). The first number having exactly k representations is A284758(k) for k >= 0 or A066460(k+1) for k > 0. - Bernard Schott, Apr 08 2017

Links

Crossrefs

Cf. A066460, A125134, A284758.

Programs

  • Mathematica
    brazBases[n_] := Select[Range[2, n - 2], Length[Union[IntegerDigits[n, #]]] == 1 &]; Table[Length[brazBases[n]], {n, 100}]
  • PARI
    a(n) = sum(i=2, n-2, #vecsort(digits(n,i), , 8)==1) \\ David A. Corneth, Apr 08 2017

A329383 Positive integers that have more Brazilian representations than any smaller positive integer.

Original entry on oeis.org

1, 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

Views

Author

Daniel Lignon, Dec 30 2019

Keywords

Comments

By analogy with highly composite numbers (A002182), these numbers could be called highly Brazilian numbers.
Also, records in A284758.
The representation n = 11_(n-1) is allowed in A066044, but it is not allowed for Brazilian numbers. Hence 3 = 11_2 = A066044(2) is not Brazilian and therefore not highly Brazilian. However, except for 3, the sequences A066044 and this one are the same.
The first time the name "highly Brazilian number" was used is in Daniel Lignon's book in reference. - Bernard Schott, Jul 27 2020

Examples

			40 is a term since 40 = 1111_3 = 55_7 = 44_9 = 22_19 and it's the smallest number with 4 representations as a Brazilian number.

References

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

Links

Crossrefs

Cf. A066044, A279930, A284758, A309039, A309493, A371812.

A066460 a(n) is the least positive integer k such that k is a repdigit number in exactly n different bases B, where 1

Original entry on oeis.org

1, 3, 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

Views

Author

Robert G. Wilson v, Jan 02 2002

Keywords

Comments

All numbers n are repdigit in base 1 and in all bases greater than n, therefore we restrict the sequence to bases between 1 and n exclusively.

Examples

			a(4) = 24 since 24_10 = 44_5 = 33_7 = 22_11 = 11_23.

Links

Crossrefs

Cf. A066044, A284758.

Programs

  • Mathematica
    rp[n_, b_] := 1 == Length@ Union@ IntegerDigits[n, b]; c[1] = c[2] = 0; c[n_] := c[n] = Block[{q = Floor@Sqrt@n}, 1 + Length@ Select[Range[2, q], rp[n, #] &] + Length@ Select[Divisors[n] - 1, q < # <= n/2 && rp[n, #] &]]; a[n_] := Block[{k = 1}, While[c[k] != n, k++]; k]; Table[a[j], {j, 0, 30}] (* Giovanni Resta, Apr 07 2017 *)

Extensions

Edited by John W. Layman, Jan 16 2002
a(0) changed to 1 by Giovanni Resta, Apr 07 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

Views

Author

Bernard Schott, Apr 12 2017

Keywords

Comments

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

Examples

			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.

References

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

Links

Crossrefs

Intersection of A002182 (highly composite) and A329383 (highly Brazilian numbers).
Cf. A284758.

Extensions

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

Views

Author

Bernard Schott, Jun 15 2017

Keywords

Comments

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.

Examples

			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.

References

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

Links

Crossrefs

Cf. A085104, A125134, A220570, A220571, A284758, A290015, A290016, A290017, A290018, A329383.

Programs

  • Mathematica
    Select[Range@ 145, Function[n, Count[Range[2, n - 2], b_ /; SameQ @@ IntegerDigits[n, b]] == 1]] (* Michael De Vlieger, Jun 16 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

Views

Author

Bernard Schott, Jul 17 2017

Keywords

Comments

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).

Examples

			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.

Links

Crossrefs

Cf. A085104, A125134, A220570, A220571, A257521, A284758, A288783, A290016, A290017, A290018, A329383.

Programs

  • Maple
    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 *)

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

Views

Author

Daniel Mondot, Apr 06 2024

Keywords

Comments

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.

Examples

			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.

Links

Crossrefs

Cf. A329383, A284758, A242397.

Programs

  • C
    #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
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.