Math Questions: A096216 & A116537

[Matthew Conroy]

On Mar 30, 2006, at 9:30 AM, Leroy Quet wrote:
> Sequence A096216 is (paraphrasing the official name of the sequence):
> a(1)=1, a(n) = the number of earlier terms of the sequence which are
> coprime to n.
> First, is A096216 such that a(2n) is always <= to both a(2n+1) and
> a(2n-1)?

No: a(106)=40>37=a(105).

Smallest n's for which a(2n)>a(2n-1) are 53,1733,2048,2468,2888,2993.
Smallest n's for which a(2n)>a(2n+1) are 1732,2467,2677,2887,3202,3217.

> Also, it SEEMS like the limits, where {a(k)} is either sequence,
>
> (1/n^2) * sum{k=1 to n} a(k), as n -> inf,
>
> approaches one of two nonzero finite constants, the constant  
> depending on
> which sequence is {a(k)}.

Let A(n) = 1/n^2*sum(k=1 to n) a(k), for a(k) the sequence A096216.
Then:
A(10) = 0.3
A(100) = 0.2653
A(1000)=0.284195
A(10000)=0.28841999
A(100000)=0.2922918699

It certainly doesn't seem to be converging too fast.  Perhaps it's not
converging at all.  Note that A(100000)-A(10000) is almost the same
size as A(10000)-A(1000).

The first 100,000 terms are in this file:
http://www.madandmoonly.com/doctormatt/mathematics/a096216rez.txt.gzip

-Matt