Sum of non divisors of n

[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.)

-- 
Franklin T. Adams-Watters
16 W. Michigan Ave.
Palatine, IL 60067
847-776-7645


__________________________________________________________________
Switch to Netscape Internet Service.
As low as $9.95 a month -- Sign up today at http://isp.netscape.com/register

Netscape. Just the Net You Need.

New! Netscape Toolbar for Internet Explorer
Search from anywhere on the Web and block those annoying pop-ups.
Download now at http://channels.netscape.com/ns/search/install.jsp