[seqfan] The anti-Carmichael pretenders

Tomasz Ordowski tomaszordowski at gmail.com
Thu Sep 13 09:42:28 CEST 2018

Dear SeqFans!

Let a(n) be the smallest k > 1 such that n^k == n (mod k)
and p-1 does not divide k-1 for every prime p dividing k,
so (k,b^k-b)=1 for some b <> n.

It seems that all terms of the sequence are semiprimes:
k = pq with primes p < q such that p-1 does not divide q-1.

The question: Is this sequence (un)bounded?

Cf. http://oeis.org/A316940 (unbounded).

Best regards,

The primary pretenders https://oeis.org/A000790 is bounded,
namely a(n) <= 561. All terms k < 561 are semiprimes k = pq
with primes p <= q such that p-1 divides q-1, thus p-1 divides k-1:
https://oeis.org/history/view?seq=A108574&v=29 (the last comment).

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