[seqfan] Are there "harmonic-Bernoulli" pseudoprimes?

[Tomasz Ordowski]

Hello SeqFans,

I noticed an interesting relationship
of the harmonic numbers H(n)
and Bernoulli numbers B(k)
"two in one" with primes.

Theorem: If p > 3 is a prime,
then (p-1)*H(p-2) == p*B(p-1) == -1 (mod p).

Conjecture: For n > 3,
(n-1)*H(n-2) == n*B(n-1) (mod n)
if and only if n is a prime.

Ami tries to find possible counterexamples:
pseudoprimes, composite numbers m such that
Numerator((m-1)*H(m-2) - m*B(m-1)) == 0 (mod m).

Has anyone already considered the above congruence?

I am asking for comments.

Regards,

Tom