

A279930


Numbers which are highly composite and highly Brazilian.


4



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
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OFFSET

1,2


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 bfile come from the new terms in bfile of A066044 found by Giovanni Resta.  Bernard Schott, Aug 03 2019


REFERENCES

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


LINKS

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


EXAMPLE

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.


CROSSREFS

Intersection of A002182 (highly composite) and A066044 (highly Brazilian numbers).
Cf. A284758.
Sequence in context: A044507 A101860 A163636 * A292891 A182464 A211571
Adjacent sequences: A279927 A279928 A279929 * A279931 A279932 A279933


KEYWORD

nonn,base


AUTHOR

Bernard Schott, Apr 12 2017


EXTENSIONS

Typo in a(18) corrected by J. Lowell, Jul 08 2019
a(29)a(35) from Bernard Schott, Jul 12 2019


STATUS

approved



