%I #34 Aug 21 2019 06:01:15

%S 0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,1,2,2,1,2,2,2,2,2,2,2,2,2,3,2,2,2,2,3,

%T 3,3,3,3,3,3,4,4,4,4,4,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,

%U 5,5,5,5,5,5,5

%N The number of primorials that neither exceed nor divide the n-th colossally abundant number.

%C Also: the number of primorials p#<m such that p#/phi(p#)>m/phi(m), where m is the n-th colossally abundant (CA) number.

%C Note that only two CA numbers are primorials (2 and 6); all other CA numbers are not squarefree, while all primorials are squarefree.

%C The sequence is not monotonic but tends to grow, albeit very slowly.

%D G. Robin, Grandes valeurs de la fonction somme des diviseurs et hypothèse de Riemann. J. Math. Pures Appl. 63 (1984), 187-213.

%H Amiram Eldar, <a href="/A213259/b213259.txt">Table of n, a(n) for n = 1..10000</a>

%H L. Alaoglu and P. Erdős, <a href="http://www.renyi.hu/~p_erdos/1944-03.pdf">On highly composite and similar numbers</a>, Trans. Amer. Math. Soc., 56 (1944), 448-469.

%H P. Erdős and J.-L. Nicolas, <a href="http://archive.numdam.org/article/BSMF_1975__103__65_0.pdf">Répartition des nombres superabondants</a>, Bull. Soc. Math. France 103 (1975), pp. 65-90.

%H J.-L. Nicolas, <a href="http://dx.doi.org/10.1016/0022-314X(83)90055-0">Petites valeurs de la fonction d'Euler</a>, J. Number Theory 17, no.3 (1983), 375-388.

%H S. Ramanujan, <a href="http://dx.doi.org/10.1023/A:1009764017495">Highly composite numbers</a>, Annotated and with a foreword by J.-L. Nicolas and G. Robin, Ramanujan J., 1 (1997), 119-153.

%F Let x be the largest prime factor of the n-th CA number m, then:

%F (1) a(n) >= [sqrt(x)/log(x)] for all nonzero terms a(n).

%F (2) a(n) < 2 sqrt(x)/log(x) for all terms a(n).

%F (3) a(n+1) <= a(n)+1 (due to the factorization pattern of CA numbers).

%F Corollary: there are only finitely many zero terms a(n).

%F All formulas can be checked directly for small x (e.g., for x<100000).

%F For large x, (1) and (2) follow from Robin's Lemma (Robin, 1984, p.190).

%e For the first five CA numbers m, each smaller primorial divides m; therefore the initial five terms are zeros.

%e The 6th CA number, 360, is not divisible by one smaller primorial, 210; thus a(6)=1.

%e The 13th CA number, 21621600, is not divisible by two smaller primorials, 510510 and 9699690; thus a(13)=2.

%Y Cf. A004490 (colossally abundant numbers), A002110 (primorials).

%K nonn

%O 1,13

%A _Alexei Kourbatov_, Jun 07 2012