A354770 Numbers k such that d(k)/log(k) sets a new record, where d(k) is the number-of-divisors function A000005(k).

2, 60, 120, 180, 240, 360, 720, 840, 1260, 1680, 2520, 5040, 7560, 10080, 15120, 20160, 25200, 27720, 50400, 55440, 83160, 110880, 166320, 221760, 277200, 332640, 498960, 554400, 665280, 720720, 1081080, 1441440, 2162160, 2882880, 3603600, 4324320, 6486480, 7207200, 8648640

Offset

1,1

Comments

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?

References

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

Links

Table of n, a(n) for n=1..39.

Example

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.

Mathematica

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 *)

Crossrefs

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

Sequence in context: A117482 A078511 A370881 * A087004 A328859 A141055

Adjacent sequences: A354767 A354768 A354769 * A354771 A354772 A354773

Keyword

nonn

Author

N. J. A. Sloane, Jun 22 2022

Extensions

More terms from Amiram Eldar, Jun 22 2022

Status

approved