Sum of non divisors of n

[f.firoozbakht at sci.ui.ac.ir]

Quoting franktaw <franktaw at netscape.net (Franklin T. Adams-Watters)>:

> f.firoozbakht at sci.ui.ac.ir wrote:
> >Quoting "y.kohmoto" <zbi74583 at boat.zero.ad.jp>:
> >>     S_3          0, 0, 0, 0, 0, 4, 0, 6, 6, 18, 0, 39,
> >>     Name : Sum{k : 1<=k<=n , k is not a divisor of n , k is not coprime to
> >> n} = n*(n+1)/2+1-Phi(n)*n/2-Sigma(n)
> >>
> >I think the sequence S_3 is interesting.
> >
> >It is obvious that if p is prime then S_3(p)=0,
>
> Yes, this is obvious.
>
> >seems that
> >4 is the only composite number n less than 10^7, such that
> >S_3(n)=0.
>
> This is also obvious.  If n has a proper factorization d*e with d > 2, then
> (d-1)*e is neither a divisor of n nor relatively prime to it.  And every
> composite number > 4 has such a factorization.
>
> >It seems that the only composite numbers n such that
> >S_3(n)< n are : 4, 6, 8 & 9
>
> And this is also obvious.  If n = d*e with d > 4, then (d-1)*e and (d-2)*e
> are two such numbers, and their sum is greater then n.  And every composite
> number greater than 9 has such a factorization.
>
> (Begin rant mode.)  This is a prime example of somebody leaning on a tool,
> instead of thinking a bit.  Try just a few values of n by hand, and it is
> immediately obvious what is happening.  Letting the computer do 10^7 values
> doesn't really tell you much.  (End rant mode.)


Hello,

I think in number theory many problems come to mind by
some computational results. Then, in the next step we
begin to prove the observations.

I only wrote my observations and left the proofs to others,
because I was too busy to prove all my observations.

Note that I sent the observations about one hour after
getting Yasutoshi's email and you sent the proofs of them
after three days.

Anyway thanks for your nice proofs.

Best wishes,

Farideh



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