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Numbers that give records for f(n)= (log(d(n))*log(log(n)))/(log(2)*log(n)) with d(n) the number of divisors.
3

%I #18 Nov 02 2012 12:36:06

%S 2,3,4,6,8,10,12,18,24,36,48,60,120,180,240,360,720,840,1260,1680,

%T 2520,5040,10080,15120,27720,55440,110880,166320,332640,720720,

%U 1441440,2162160,4324320,21621600,367567200,6983776800

%N Numbers that give records for f(n)= (log(d(n))*log(log(n)))/(log(2)*log(n)) with d(n) the number of divisors.

%C It is proved that the function f reaches its maximum for n = 6983776800, and that max n>=2 f(n) = 1.5379. The proof deals with superior highly composite numbers introduced by Ramanujan. So n = 6983776800 is the final term of this sequence.

%H J. L. Nicolas and G. Robin, <a href="http://dx.doi.org/10.4153/CMB-1983-078-5">Majorations explicites pour le nombre de diviseurs de N</a>, Canad. Math. Bull. 26(1983), pp. 485-492.

%o (PARI) f(n) = {maxx = -999; for (i=2, n, x = (log(numdiv(i))*log(log(i)))/(log(2)*log(i)); if (x > maxx, maxx = x;print1(i, ",");););}

%K nonn,fini,full

%O 1,1

%A _Michel Marcus_, Nov 01 2012

%E a(36) and keyword "full" added by _Donovan Johnson_, Nov 01 2012