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A210594 "Factor-dense" numbers: integers n where (# of proper divisors of n) / log(n) sets a new record. 1
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 (list; graph; refs; listen; history; text; internal format)
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.

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

(Python 3)

import math

def factors(n):

.result = {1}

.for d in range(2, round(math.sqrt(n)) + 1):

..if n % d == 0:

...result |= {d, n // d}

.return result

def factor_density(n):

.return len(factors(n)) / math.log(n)

n = 2

max_fd = -1

while True:

.fd = factor_density(n)

.if fd > max_fd:

..print(n)

..max_fd = fd

.n += 1

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

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Last modified November 20 10:09 EST 2019. Contains 329334 sequences. (Running on oeis4.)