A217660 Numbers that give records for f(n)= (log(d(n))*log(log(n)))/(log(2)*log(n)) with d(n) the number of divisors.

2, 3, 4, 6, 8, 10, 12, 18, 24, 36, 48, 60, 120, 180, 240, 360, 720, 840, 1260, 1680, 2520, 5040, 10080, 15120, 27720, 55440, 110880, 166320, 332640, 720720, 1441440, 2162160, 4324320, 21621600, 367567200, 6983776800

Offset: 1

Author

Michel Marcus, Nov 01 2012

Keywords

• nonn
• fini
• full

Comments

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.

Links

• J. L. Nicolas and G. Robin, Majorations explicites pour le nombre de diviseurs de N, Canad. Math. Bull. 26(1983), pp. 485-492.

Programs

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, ",");););}

Extensions

a(36) and keyword "full" added by Donovan Johnson, Nov 01 2012