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

[Tomasz Ordowski]

Are there odd composite numbers n such that
2^{n-1} == 1 (mod n D_{n-1} / 2} with gcd(n, D_{n-1}) > 1 ?
Thanks to Amiram for finding the numbers I asked for:
23872213, 36974341, 427294141, ...
Its data extension (as a proper subset of A331021) seems to be right:
23872213, 36974341, 427294141, 526359289, 550230409, 2129304997,
3196344061, 3598497631, 6913548721, 16267168711, 24449995753, 25546873717,
28751571577, 41521427221, ...

By the way, let's ask:
Are there odd numbers n > 1 such that p^{n-1} == 1 (mod n) for every prime
p | D_{n-1} ?
These are numbers n > 1 such that p^{n-1} == 1 (mod n) for every prime p
with p-1 | n-1.
A proper subset of A316907.
Amiram Eldar:
Well, I checked all the 10^4 terms of b316907 (the largest is ~2.5*10^11)
and found none.
So perhaps such numbers do not exist at all.

T. Ordowski

pon., 6 wrz 2021 o 19:52 Tomasz Ordowski <tomaszordowski at gmail.com>
napisał(a):

> 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>
>>
>>