[seqfan] Re: Anti-Carmichael numbers

[Ami Eldar]

Anti-Lucas-Carmichael numbers: 4, 6, 9, 10, 12, 16, 18, 21, 22, 24, 25, 28,
30, 33, 34, 36, 40, 42, 45, 46, 48, 49, 52, 54, 55, 57, 58, 60, 64, 66, 69,
70, 72, 76, 77,
78, 81, 82, 84, 85, 88, 90, 91, 93, 94, 96, 100, ...
They are analogous to A121707 as A006972 is analogous to A002997.

The are also numbers that are both Anti-Carmichael and
Anti-Lucas-Carmichael: 55, 77, 115, 161, 187, 203, 209, 221, 235, 247, 253,
295, 299, ..
They are analogous to Williams numbers (numbers that are both Carmichael
and Lucas-Carmichael).

None of these sequences are in OEIS. The last one (Williams numbers) is not
there since there is no known example...

Ami





On Fri, Jul 20, 2018 at 11:38 AM, Tomasz Ordowski <tomaszordowski at gmail.com>
wrote:

> Dear Ami,
>
> thank you for your interest in the topic.
>
> Are your anti-Lucas-Carmichael numbers in the OEIS?
>
> Can you give me the first few such numbers?
>
> Maybe their density D = 1/zeta(2) = 6/pi^2.
>
> Best regards,
>
> Thomas
>
> P.S. By the way: https://oeis.org/draft/A121707
>
>
> 2018-07-20 0:24 GMT+02:00 Ami Eldar <amiram.eldar at gmail.com>:
>
> > What about the Anti-Lucas-Carmichael numbers: numbers n such that p+1
> non |
> > n+1 for every prime p | n.
> > It seems that their density is D > 0.6.
> >
> > On Thu, Jul 19, 2018 at 2:48 PM, Tomasz Ordowski <
> tomaszordowski at gmail.com
> > >
> > wrote:
> >
> > > Dear SeqFan!
> > >
> > > See http://oeis.org/A121707.
> > >
> > > Here's a simple definition of these numbers:
> > >
> > > Numbers n such that p-1 does not divide n-1 for every prime p dividing
> n.
> > >
> > > PROOF.
> > >
> > > Carmichael numbers: composite n coprime to
> 1^(n-1)+2^(n-1)+...+(n-1)^(n-
> > > 1).
> > >
> > > Anti-Carmichael numbers: n such that p-1 non | n-1 for every prime p |
> n.
> > >
> > >  Odd numbers n>1 such that n | 1^(n-1)+2^(n-1)+...+(n-1)^(n-1). (**)
> > >
> > > Equivalently: Numbers n>1 such that n^3 | 1^n+2^n+...+(n-1)^n. (*)
> > >
> > > Cf. http://oeis.org/A121707 (see conjecture in the second comment).
> > >
> > > Professor Schinzel (2015) proved the equivalence (*) <=> (**). QED.
> > >
> > > Problem: What is the natural density D of the set of these numbers?
> > >
> > > Can be proved that 0.1 < D < 0.3.  My conjecture: D = 2/pi^2.
> > >
> > > Sincerely,
> > >
> > > Tomasz Ordowski
> > > ______________
> > > Szanowny Panie,
> > > Podzielność (*) n^3|1^n+...+(n-1)^n=S_n(n) wymaga żeby n było
> > nieparzyste,
> > > bowiem z 2^a|n (a>0) wynika S_n(n)= n/2 mod 2^(a+2). Zatem (*) jest
> > > rownoważna  podzielności n^3|(1^n+(n-1)^n)+(2^n+(n-2)^
> > > n)+...+((n-1)^n+1^n).
> > > Ale przy n nieparzystym dla każdego i :i^n+(n-i)^n=n^2*i^(n-1), zatem
> > > podzielność (*) jest rownoważna podzielności (**) n|S_(n-1)(n). Z
> drugiej
> > > strony podzielność (**) i warunek 4 nie dzieli n wymagają
> nieparzystości
> > n.
> > > Lącze pozdrowienia.
> > >                 Andrzej Schinzel
> > >
> > > --
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> > >
> >
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> >
>
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