# Sum Over Coprime Integers = 0 (or 1)

Michele Dondi blazar at pcteor1.mi.infn.it
Fri Sep 24 10:48:28 CEST 2004

```On Tue, 10 Aug 2004, Leroy Quet wrote:

> 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
>

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```