Sum Over Coprime Integers = 0 (or 1)

[Leroy Quet]

I have sent the following copied message to seqfan, but it has not been 
sent back to my email address for some reason. I have posted some new 
info below as a follow-up.



>Let the sequence {a(k)} be such that
>
>a(1) = 1;
>
>For each m >= 3,
>
>0 = sum{1<=k<m,GCD(k,m)=1} a(k).
>
>I get, by hand, the sequence beginning:
>
>1, -1, -1, 1, -1, 1, 1, -1, -1,...
>
>
>Is there a closed form for this sequence?
>
>It is doubtful, especially since I only have 10 terms above, but could it 
>be that every term is either 1 or -1?
>(I bet a counterexample, if one exists, is EASILY found.)
>:)
>
>thanks,
>Leroy Quet 



I just posted the following to the EIS, with more terms calculated 
(correctly??).

%I A000001
%S A000001 1,-1,-1,1,-1,1,1,-1,-1,1,-1,1,3,-3,-1,1,-3
%N A000001 a(1)=1; for each n >= 3, sum{1<=k<n,GCD(k,n)=1} a(k) 
= 0.
%e A000001 Since the positive integers < 8 and coprime with 8 are 1, 
3, 5, 7,
we have a(1) +a(3) +a(5) +a(7) = 1 -1 -1 +1 = 0.
%O A000001 1
%K A000001 ,more,sign,
%A A000001 Leroy Quet (qq-quet at mindspring.com), Aug 10 2004

As you can see, there is a 3 and some -3's, so each term is not 
necessarily -1 or 1.

thanks,
Leroy Quet