[seqfan] Bernoulli harmonic numbers

[Tomasz Ordowski]

Dear readers,

I have defined a sequence of interesting fractions.

Let F(n) = Sum_{k=1..n} (B_{k-1}+1/k) = N(n)/D(n), for n > 0.

2/1, 2/1, 5/2, 11/4, 35/12, 37/12, 13/4, 27/8, 1243/360, 1279/360,
1339/360, 1369/360, 9139/2520, 9319/2520, 12427/2520, 25169/5040, ...

I noticed (provable) dependencies:
(*) If p > 3 is prime, then F(p-2) == 0 (mod p), i.e. p | N(p-2).
(**) If p > 2 is prime, then F(p-1) == -1 (mod p), i.e. p | N(p-1)+D(p-1).
Are these congruences (*) and (**) equivalent by the following equality?
F(2n) = F(2n-1) + 1/(2n), for n > 1.

Conjectures:
(*) For n > 3, n | N(n-2) if and only if n is prime.
(**) For n > 2, n | N(n-1)+D(n-1) if and only if n is prime.
Question: How to prove that such pseudoprimes n do not exist?
If there is such a composite n, does it have to be a Carmichael number?

Finally, from Amiram Eldar:
(*) Primes p such that p^2 | N(p-2) are 17, 2663, 10589, ...
(**) Primes p such that p^2 | N(p-1)+D(p-1) are 7, 1181, ...

Sincerely,

Thomas Ordowski
_____________
https://oeis.org/A174341 / https://oeis.org/A174342
For n > 0, F(n) = -1 + Sum_{k=0..n-1} A174341(k)/A174342(k).