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%I #37 Sep 15 2022 16:34:37
%S 2,6,12,24,36,48,60,120,180,240,360,720,840,1260,1680,2520,5040,7560,
%T 10080,15120,20160,25200,27720,50400,55440,83160,110880,166320,221760,
%U 277200,332640,498960,554400,665280,720720,1081080,1441440,2162160,2882880,3603600
%N "Factor-dense" numbers: integers n where (# of proper divisors of n) / log(n) sets a new record.
%C Let d(n) = the number of proper divisors of n (A032741).
%C Define the "factor density" of n as f(n) = d(n) / log(n).
%C n is "factor dense" if f(m) < f(n) for all integers m where 2 <= m < n.
%C Missing highly-composite numbers (A002182) are 4 and 45360.
%C 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
%H Robert G. Wilson v, <a href="/A210594/b210594.txt">Table of n, a(n) for n = 1..102</a>
%t 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 *)
%o (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
%Y Cf. A189686.
%K nonn
%O 1,1
%A _Daniel Bishop_, Mar 23 2012
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