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

Author

Daniel Bishop, Mar 23 2012

Keywords

• nonn

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

Crossrefs

Cf. A189686.

Programs

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 *)

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