Number-Divisors Almost = ln(m) + 2c-1
Leroy Quet
qq-quet at mindspring.com
Tue Dec 2 03:18:11 CET 2003
In the sci.math thread (posted about a month back) "two
number-theoretical limits
(& bonus sum)":
http://groups.google.com/groups?hl=en&lr=&ie=UTF-8&safe=off&threadm=b4be2fd
f.0310301645.38b9a167%40posting.google.com&rnum=6&prev=
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))|
thanks,
Leroy Quet
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