Number-Divisors Almost = ln(m) + 2c-1

Leroy Quet qq-quet at
Tue Dec 2 03:18:11 CET 2003

In the sci.math thread (posted about a month back) "two 
number-theoretical limits 
(& bonus sum)":

I figured (and Martin Cohen confirmed {referencing Hardy & Wright theorem 
320}) that d(m), the number of positive divisors of m, was such that

limit{n-> infinity} (1/n) (sum{m=1 to n} d(m))   - ln(n)

= 2*c - 1, where c is Euler's constant (.5772...).

So, I am wondering,

what is the sequence of increasing positive integers where:

a(1) = 1;

|d(a(m)) - ln(a(m)) +1 -2c|  <

|d(a(m-1)) - ln(a(m-1)) +1 -2c|

for all m >= 2 ? 

In other words, what is the sequence of positive integers where the 
number of divisors in each term better approximates the asymptotical 
appoximation of the 'average number of divisors' than any previous term?

We could also ask about the simplier defined b-sequence:

b(1) = 1;

|d(b(m)) - ln(b(m))|  <

|d(b(m-1)) - ln(b(m-1))|

Leroy Quet 

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