[seqfan] Re: Let p_1 .. p_k be prime divisors of n counted with multiplicity, then consider the rational number (p_1-1)/p_1 + (p_2-1)/p_2 + ... (p_k-1)/p_k

[M. F. Hasler]

Thomas,

the main "lines" or "rays" correspond simply to the lines you get for  n =
m p
where m >= 1 is some small factor and p runs over all primes :

m=1 : n = p, the primes:
s(n)  =  (p-1)/p  =>  D(N) = N-1,  i.e.,  y ~ x

m=2 : n = 2p : even semiprimes:
s(n)  =  1/2 + (p-1)/p   = (3p/2-1/2) / p
=>  D(N) = 3N/2 - 1/2,   i.e.,  y ~ 3/2 x,

and similarly :
m=3  =>  y ~ 5/3 x
m=4  =>  y ~ 2 x
m=5  => y  ~  9/5,
etc.:
You can easily figure out the expression and see to what asymptote
corresponds any family  s(m*p).

- Maximilian


On Fri, Mar 11, 2022 at 11:36 AM Jakob Schulz <bruderjakob17 at gmail.com>
wrote:

>  Dear Thomas,
>
> While I cannot answer any of your questions, I think that D(n) is already
> in the OEIS as A083346(n). This is because s(n) = (1 - 1/p_1) + ... + (1 -
> 1/p_k) = k - (1/p_1 + ... + 1/p_k) = k - r(n), where r(n) is defined as in
> the OEIS-entry.
>
> Best,
> Jakob
>
> Am Fr., 11. März 2022 um 08:29 Uhr schrieb Thomas Scheuerle via SeqFan <
> seqfan at list.seqfan.eu>:
>
> > Hi,
> >
> > If we see the numerators and the denominators of
> > s(n) = (p_1-1)/p_1 + (p_2-1)/p_2 + ... (p_k-1)/p_k
> > as integersequences N(n), D(n)
> > then we will observe mysterious lines and other more
> > complicated looking structures,if we plot N(n) against D(n) in a X,Y
> > scatter plot.
> > This fact was enough motivation for me,
> > but now I am asking my self what other interesting properties
> > or maybe even applications could such a sequence have.
> >
> > Maybe someone out there has some ideas what properties should be checked
> > or what properties could be possibly expected from such a sequence ?
> > (I should better say sequences as we have numerator and denominator
> here.)
> >
> > best rgeards
> >
> > Thomas
> >
> >
> >
> >
> > --
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> >
>
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