A210594 "Factor-dense" numbers: integers n where (# of proper divisors of n) / log(n) sets a new record.

2, 6, 12, 24, 36, 48, 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

Offset

1,1

Comments

Let d(n) = the number of proper divisors of n (A032741).

Define the "factor density" of n as f(n) = d(n) / log(n).

n is "factor dense" if f(m) < f(n) for all integers m where 2 <= m < n.

Missing highly-composite numbers (A002182) are 4 and 45360.

An alternative definition of factor density is g(n) = tau(n) / log(1+n), where tau(n) is the total number of divisors of n (A000005). Then records for g(n) appear to be set at all members of this sequence together with 1 and 4. - Hal M. Switkay, Sep 07 2022

Links

Robert G. Wilson v, Table of n, a(n) for n = 1..102

Mathematica

f[n_] := N[(DivisorSigma[0, n] - 1)/Log[n]]; mx = 0; lst = {}; Do[ If[ f[n] > mx, mx = f[n]; AppendTo[lst, n]], {n, 2, 4000000, 2}]; t (* T. D. Noe, Mar 23 2012 *)

Prog

(PARI) lista(nn) = {my(m=0); for (n=2, nn, my(mm = (numdiv(n)-1)/log(n)); if (mm > m, print1(n, ", "); m = mm); ); } \\ Michel Marcus, Sep 08 2022

Crossrefs

Cf. A189686.

Sequence in context: A099993 A011778 A067718 * A307252 A306625 A262986

Adjacent sequences: A210591 A210592 A210593 * A210595 A210596 A210597

Keyword

nonn

Author

Daniel Bishop, Mar 23 2012

Status

approved