[seqfan] Smallest witnesses

[Tomasz Ordowski]

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




<#m_5771181396519008915_m_-1684118368193253876_m_-5719339244750301007_m_2325595904686261964_m_4217233533039576782_DAB4FAD8-2DD7-40BB-A1B8-4E2AA1F9FDF2>