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 1-4 of 4 results.

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

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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

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

Original entry on oeis.org

7, 15, 40, 336, 1440, 5405400
Offset: 1

Views

Author

Bernard Schott, Aug 04 2019

Keywords

Comments

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.

Examples

			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.

Links

Crossrefs

Cf. A002182, A279930, A309039, A329383.
Showing 1-4 of 4 results.

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