A279930 Numbers which are highly composite and highly Brazilian.

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

Author

Bernard Schott, Apr 12 2017

Keywords

• nonn
• base

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

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

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