A354770 - OEIS

%I #23 Sep 22 2022 01:40:09

%S 2,60,120,180,240,360,720,840,1260,1680,2520,5040,7560,10080,15120,

%T 20160,25200,27720,50400,55440,83160,110880,166320,221760,277200,

%U 332640,498960,554400,665280,720720,1081080,1441440,2162160,2882880,3603600,4324320,6486480,7207200,8648640

%N Numbers k such that d(k)/log(k) sets a new record, where d(k) is the number-of-divisors function A000005(k).

%C A related sequence, not yet in the OEIS, is "Numbers k such that log(d(k))/log(k) > log(d(m))/log(m) for all m > k".  It begins 2, 4, 6, 12, 24, 36, 60, 72, 120, 180, 240, 360, 420, 720, 840, 1260, 1680, 2520, 5040, 7560, ..., and up to this point it agrees with A236021 (except that it doesn't include 1). Does it continue to agree with A236021?

%D David desJardins, Posting to Math Fun Mailing List, Jun 22 2022.

%e The values of d(k)/log(k) for k = 2, 3, ... are 2.885390082, 1.820478453, 2.164042562, 1.242669869, 2.232442506, 1.027796685, 1.923593388, 1.365358840, 1.737177928, 0.8340647828, ... and reach record highs at k = 2 (2.885390082...), k = 60 (2.930872040...), and so on.

%t s = {}; rm = 0; Do[If[(r = DivisorSigma[0, n]/Log[n]) > rm, rm = r; AppendTo[s, n]], {n, 2, 10^5}]; s (* _Amiram Eldar_, Jun 22 2022 *)

%Y Cf. A000005, A066523, A236020, A236021, A354768, A354369.

%K nonn

%O 1,1

%A _N. J. A. Sloane_, Jun 22 2022

%E More terms from _Amiram Eldar_, Jun 22 2022