[seqfan] Bernoulli numbers and new fractions in relation to primes and Carmichael numbers

[Tomasz Ordowski]

Dear SeqFans,

I defined the sequence of fractions F(n):

2/1, 0/1, 1/1, 1/16, 1/1, 1/36, 1/1, 1/64, 7/27, 1/100, 1/1, 1/144, -37/1,
1/196, 37/75, 1/256, -211/1, 1/324, 2311/1, 1/400, -407389/49, ...

F(n) = N(n-1) / n + D(n-1) / n^2, where the k-th Bernoulli number B(k) =
N(k) / D(k).
Cf. https://en.wikipedia.org/wiki/Bernoulli_number#Related_sequences

It seems that the numerator of F(n) is the numerator of (B(n-1) + 1/n).
Cf. https://oeis.org/history/view?seq=A174341&v=28 (see my comment).
However, the denominator of F(n) is generally different from A174342(n-1).

Let a(n) be the denominator of F(n) = N(n-1)/n+D(n-1)/n^2, where B(k) =
N(k)/D(k), as above.

a(n) = 1, 1, 1, 16, 1, 36, 1, 64, 27, 100, 1, 144, 1, 196, 75, 256, 1, 324,
1, 400, 49, 484, 1, 576, 125, 676, 243, 784, 1, 900, 1, 1024, 363, 1156,
1225, 1296, 1, 1444, 169, 1600, 1, 1764, 1, 1936, 135, 2116, 1, 2304, 343,
2500, 867, 2704, 1, 2916, 3025, 3136, 361, 3364, 1, 3600, 1, 3844, 1323,
4096, 845, 4356, 1, 4624, 1587, 4900, 1, 5184, 1, 5476, 625, 5776, 5929,
6084, 1, 6400, 2187, 6724, 1, 7056, 1445, 7396, 2523, 7744, 1, 8100, 1183,
8464, 961, 8836, 9025, 9216, 1, 9604, 3267, 10000, ...

Conjecture: For n > 1, a(n) = 1 if and only if n is prime.
Question: Is this equivalent to the Agoh-Giuga conjecture?
https://en.wikipedia.org/wiki/Agoh%E2%80%93Giuga_conjecture
In other words: For n > 1, n^2 | n N(n-1) + D(n-1) if and only if n is
prime.
Jonathan Sondow proved this in one direction using the von Staudt and
Clausen theorem:
http://mathworld.wolfram.com/vonStaudt-ClausenTheorem.html

Note: N(p-1) == - D(p-1) / p (mod p^2) for primes p = 2 and p = 1277.
Also, Sum_{k=1..p-1} k^{p-1} == -1 (mod p^2) for the prime p = 1277.

In relation to the Carmichael numbers:
(*) If n is a Carmichael number, then a(n) <= n.
(**) For n > 1, if a(n) = n, then n is a Carmichael number.
Carmichael numbers n for which a(n) < n are 561, 1105, 46657, 52633,
188461, 670033, 825265, 838201, ... Data from Amiram Eldar.
Composite numbers n for which a(n) < n, that are not Carmichael numbers,
are 5005, 28405, 47125, ... Found by Amiram Eldar.
New Conjecture: For a composite n, a(n) <= n and a(n) is squarefree if and
only if n is a Carmichael number.

Are these statements (*) and (**) provable?

I am asking for comments.

Best regards,

Thomas Ordowski
_______________
Bernoulli numbers: https://oeis.org/A027641 / https://oeis.org/A027642
B(k) = N(k) / D(k).
As is well known, if p is prime, then p | D(p-1). https://oeis.org/A000040
Composite numbers m such that m | D(m-1) are the Carmichael numbers:
https://oeis.org/A002997 (see penultimate comment).