Re: [math-fun] highly composite numbers

OK, I did a computation. Letting d(N) denote the number of divisors of N, I find 1. for 0<N<e^380 that d(N)<2.44*2^Li(lnN) with tightness only for d(6)=4. Does this upper bound work forever? No. In an excellent example of misleading behavior for small N, I found d(N)>1591396267*2^Li(lnN) for a certain N with lnN =approx= 1000010.9002377. 2. All highly composite numbers N with 1<N<e^6969 obey d(N)>1.12*2^Li(lnN), with tightness only for d(45360)=100. This supports my conjecture that d(N)>=2^Li(lnN) for all N>0 (which is tight when N=1).