login
Natural numbers k divisible by all natural numbers < log(k) + log(1 + log(k)).
0

%I #13 Jun 16 2023 09:00:29

%S 1,2,3,4,6,12,24,60,120,420,840,2520,5040,27720,360360,720720,1441440,

%T 232792560,80313433200,144403552893600,288807105787200,

%U 5342931457063200,10685862914126400,16028794371189600,164249358725037825439200,328498717450075650878400,492748076175113476317600,656997434900151301756800

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

%C 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.

%e 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.

%e 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.

%p 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

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

%Y Cf. A003418.

%K nonn

%O 1,2

%A _Hal M. Switkay_, Jun 14 2023