[seqfan] Re: Carmichael numbers of the forms D_{2k}/2 and D_{2k}/6

[Tomasz Ordowski]

P.S. Important correction from Carl Pomerance:

Safe primes p have D_{p-1} = 6p, but not necessarily
the other way round.  For example, p = 239 works but
isn't safe.  It's not been proved that there are
infinitely many safe primes, but as mentioned, Wagstaff
and I proved there is a positive proportion of primes p
with D_{p-1} = 6p.

Thanks!

pon., 10 maj 2021 o 15:41 Tomasz Ordowski <tomaszordowski at gmail.com>
napisał(a):

> Dear readers!
>
> Let D_k be the denominator of Bernoulli number B_k.
>
>   Problem:
> Are there numbers n > 3 such that D_{n-1} = 2n ?
>   Equivalently, Product_{p prime, p-1|n-1} p = 2n.
> If so, it must be a Carmichael number divisible by 3.
> Amiram Eldar found nothing below 2^64
> in https://oeis.org/A258801
>
>   Carl Pomerance:
> I don't know the answer to this, but here's a variation:
> Are there Carmichael numbers n where D_{n-1} = 6n?
> If so, are there infinitely many?
> In a paper I'm working on with Sam Wagstaff, we show
> there is a positive proportion of primes p with D_{p-1} = 6p.
> [Note that these are Safe primes: https://oeis.org/A005385]
>
> Generally, composites n such that D_{n-1} = 6n.
> 310049210890163447, 18220439770979212619, ...
> Two Carmichael numbers found by Amiram
> (of course, they cannot be divisible by 3).
>
> Best regards,
>
> Thomas
> _____________
> https://oeis.org/A002997
> https://oeis.org/A027642
>
>