[seqfan] Re: list of all non-prime coprimes of the full set of primorial numbers
Vladimir Shevelev
shevelev at bgu.ac.il
Wed Apr 26 12:40:44 CEST 2017
Hi Jamie,
By your calculations, we have
For P(2), 3-1,
For P(3), (5-1)*(3-1),
For P(4), (7-1)*(5-1)*(3-1),
For P(5), (11-1)*(7-1)*(5-1)*(3-1), etc.
So, for P(n), Prod_{p<=prime(n)}(p-1)=
Prod_{p<=prime(n)} p * Prod_{p<=prime(n)}(1-1/p)=
e^{prime(n)+o(prime(n))-gamma}/log(prime(n)),
where gamma is Euler constant=0.5772157...
I think that the number of coprimes that are
duplicates and/or prime numbers is o(prime(n)),
since P(n)/P(n-1) is prime(n).
Therefore, it is similar that for P(n) only we have
e^{prime(n)+o(prime(n))}/log(prime(n)).
Best,
Vladimir
________________________________________
From: SeqFan [seqfan-bounces at list.seqfan.eu] on behalf of Jamie Morken [jmorken at shaw.ca]
Sent: 26 April 2017 07:48
To: Sequence Fanatics Discussion list
Subject: [seqfan] Re: list of all non-prime coprimes of the full set of primorial numbers
Hi,
For the asymptotic growth, here is the count of coprimes of each Pn.Primorial coprimes
Pn(2)=6 2
Pn(3)=30 8=((5-1)*2)
Pn(4)=210 48=((7-1)*8)
Pn(5)=2310 480=((11-1)*48)Pn(6)=30030 5760=((13-1)*480)Pn(7)=510510 92160=((17-1)*5760)That can calculate the count of coprimes using the prime numbers - 1, ie Pn(7) has 92160 coprimes < 510510but doesn't show if the coprimes are duplicates and/or prime numbers. I guess the prime number contribution
could be subtracted out, so the only unknown would be the count of duplicate coprimes then. Thanks.cheers,Jamie----- Original Message -----
From: Charles Greathouse <charles.greathouse at case.edu>
To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
Sent: Tue, 25 Apr 2017 14:56:25 -0600 (MDT)
Subject: [seqfan] Re: list of all non-prime coprimes of the full set of primorial numbers
I think it would be reasonable to re-submit the clarified sequence.
Possible name: numbers n < P coprime to P for some primorial P. Question:
what is the asymptotic growth if this sequence? Clearly a(n) > kn for all k
and all large n.
On Fri, Apr 21, 2017 at 6:43 AM Jamie Morken <jmorken at shaw.ca> wrote:
> Hi,>The definition is not clear. Note that gcd(121,2310)=11, so why is
> >121 on the list?
> >Or, if 121 is on the list, why not 49?
>
> 121 is on the list as gcd(121,210)=1, the sequence includes all coprimes
> as long as they are coprime
> to at least one primorial x. 49 isn't on the list as I only included
> coprimes of primorials if they
> are smaller than the primorial. 49 isn't coprime to any primorial greater
> than Pn(3) = 30 so it isn't on
> the list.>Also, what is Pn? What is the difference between Pn(210) and 210
> ? >Did you mean to say >Pn(4) = 210 rather than Pn(210)? >Best regards
> >NeilYes thanks I meant to say Pn(4) = 210, where 210 is the fifth
> primorial, and Pn(0) = 1
> >Best regards
> >Neil
>
>
> reply to Franklin:
> >The full set of primorials is infinite, and only 1 is coprime to all of
> them. This is just the numbers coprime to 210, which is rather arbitrary.
>
> The values that are coprime to 210 also contains 253 which isn't in the
> list (see updated definition of coprimes smaller than Pn only included in
> the sequence).
>
> Should I resubmit the sequence if the definition is correct now? Thanks.
>
> cheers,
> Jamie
>
>
>
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
>
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Charles Greathouse
Case Western Reserve University
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