A363670 Natural numbers k divisible by all natural numbers < log(k) + log(1 + log(k)).

1, 2, 3, 4, 6, 12, 24, 60, 120, 420, 840, 2520, 5040, 27720, 360360, 720720, 1441440, 232792560, 80313433200, 144403552893600, 288807105787200, 5342931457063200, 10685862914126400, 16028794371189600, 164249358725037825439200, 328498717450075650878400, 492748076175113476317600, 656997434900151301756800

Offset

1,2

Comments

The required degree of divisibility for inclusion in this sequence seems close to the maximum possible degree of divisibility. If the required degree of divisibility is increased merely by one (that is, if k is required to be divisible by all natural numbers < 1 + log(k) + log(1 + log(k))), there appear to be only these solutions: 1, 2, 6, 12, and 60. This is related to the growth rate of lcm{1,...,n} = A003418(n); see comments on A003418 by Beedassy and Sondow.

Links

Table of n, a(n) for n=1..28.

Example

60 is a term of this sequence, because log(60) + log(1 + log(60)) ~ 5.72, and 60 is divisible by all natural numbers < 5.72.
However, 180 is not a term of this sequence, because log(180) + log(1 + log(180)) ~ 7.02, and 180 is not divisible by all natural numbers < 7.02.

Maple

select(k -> andmap(d -> irem(k, d) = 0, [seq(1..floor(log(k) + log(1 + log(k))))]), [seq(1..2520)]); # Peter Luschny, Jun 14 2023

Prog

(PARI) isok(k) = for(i=1, log(k) + log(1 + log(k)), if (k % i, return(0))); 1; \\ Michel Marcus, Jun 14 2023

Crossrefs

Cf. A003418.

Sequence in context: A324178 A214570 A306348 * A078495 A161701 A038504

Adjacent sequences: A363667 A363668 A363669 * A363671 A363672 A363673

Keyword

nonn

Author

Hal M. Switkay, Jun 14 2023

Status

approved