[seqfan] Re: Smallest witnesses

Tomasz Ordowski tomaszordowski at gmail.com
Mon Nov 19 11:07:52 CET 2018

```P.S. See:

https://oeis.org/history/view?seq=A321790&v=15

https://oeis.org/draft/A321790 (my draft).

Thomas Ordowski

niedz., 18 lis 2018 o 10:25 Tomasz Ordowski <tomaszordowski at gmail.com>
napisał(a):

> Dear SeqFans!
>
> In general, let's define the sequence:
>
> a(n) is the smallest base a > 0 such that a^(n-1) == 1 (mod n)
> if and only if n is not composite, for n = 1, 2, 3, 4, ...
>
> a(n) : 1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, ... All terms are noncomposite.
>
> In particular, the subsequence, defined:
>
> b(n) is the smallest base b > 0 such that b^(k-1) =/= 1 (mod k),
> where k = A001567(n) it is n-th Fermat pseudoprime to base 2.
>
> b(n) : 3, 3, 3, 5, 3, 7, 3, 3, 5, 5, 7, 3, ... Each b(n) is an odd prime.
>
> Note that if k is a Carmichael number, then b(n) = lpf(k).
>
> Conjecture: if k is semiprime, then b(n) < lpf(k).
>
> Cf. A083876 and A285549.
>
> Best regards,
>
> Thomas
> ____________________
> https://oeis.org/A083876
> https://oeis.org/A285549
>
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