[seqfan] Re: Wieferich primes, anti-Carmichael pseudoprimes, and ?

[Ami Eldar]

Hello,

Odd composite numbers n such that 2^n == 2 (mod n D_{n-1}) and gcd(n,
D_{n-1}) > 1 do exist.
The first three are 23872213, 36974341 and 427294141.

These numbers are terms of A331021.
Assuming that they are all terms of A331021, the terms are
23872213, 36974341, 427294141, 526359289, 550230409, 2129304997,
3196344061, 3598497631, 6913548721, 16267168711, 24449995753, 25546873717,
28751571577, 41521427221, ...

Best regards,
Amiram



On Tue, Sep 7, 2021 at 9:43 AM Tomasz Ordowski <tomaszordowski at gmail.com>
wrote:

> P.S. My question can be formulated directly:
>
> Are there odd composite numbers n such that
> 2^n == 2 (mod n D_{n-1}) and gcd(n, D_{n-1}) > 1 ?
>
>
> pon., 6 wrz 2021 o 09:09 Tomasz Ordowski <tomaszordowski at gmail.com>
> napisał(a):
>
> > Hello SeqFans!
> >
> > Let's define:
> > Odd numbers n > 1 such that 2^{n-1} == 1 (mod n D_{n-1} / 2),
> > where D_k is the denominator of Bernoulli number B_k.
> >
> > Such prime numbers are Wieferich primes A001220.
> > Are there such composite numbers that are not in A316907 ?
> > A316907 (anti-Carmichael pseudoprimes) is a proper subset.
> >
> > For odd n > 1, D_{n-1} = Product_{p prime, p-1 | n-1} p.
> > Note that 2^{n-1} == 1 (mod D_{n-1} / 2) for every odd n > 1.
> >
> > Best regards,
> >
> > Thomas
> > _____________
> > A001220 - OEIS <https://oeis.org/A001220>
> > A316907 - OEIS <https://oeis.org/A316907>
> >
> >
>
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