A213259 The number of primorials that neither exceed nor divide the n-th colossally abundant number.

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, 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, 5, 5, 5, 5, 5, 5, 5

Offset

1,13

Comments

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.

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

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

References

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

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

L. Alaoglu and P. Erdős, On highly composite and similar numbers, Trans. Amer. Math. Soc., 56 (1944), 448-469.

P. Erdős and J.-L. Nicolas, Répartition des nombres superabondants, Bull. Soc. Math. France 103 (1975), pp. 65-90.

J.-L. Nicolas, Petites valeurs de la fonction d'Euler, J. Number Theory 17, no.3 (1983), 375-388.

S. Ramanujan, Highly composite numbers, Annotated and with a foreword by J.-L. Nicolas and G. Robin, Ramanujan J., 1 (1997), 119-153.

Formula

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

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

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

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

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

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

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

Example

For the first five CA numbers m, each smaller primorial divides m; therefore the initial five terms are zeros.
The 6th CA number, 360, is not divisible by one smaller primorial, 210; thus a(6)=1.
The 13th CA number, 21621600, is not divisible by two smaller primorials, 510510 and 9699690; thus a(13)=2.

Crossrefs

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

Sequence in context: A189225 A169988 A358052 * A067597 A139394 A240128

Adjacent sequences: A213256 A213257 A213258 * A213260 A213261 A213262

Keyword

nonn

Author

Alexei Kourbatov, Jun 07 2012

Status

approved