[seqfan] Fermat fraction series

[Tomasz Ordowski]

Dear readers,

I defined the series of Fermat fractions, F(n) = Sum_{k=1..n}
(2^{k-1}-1)/k, as I wrote before;
0/1, 1/2, 3/2, 13/4, 25/4, 137/12, 245/12, 871/24, 517/8, 4629/40, 8349/40,
45517/120, ...
And then I noticed that if p > 3 is a prime, then F(p) == 0 (mod p^2)
[sic].
Question to mathematician: Does this result from Wolstenholme's theorem?
A positive answer is supported by the confirmed fact that primes p for which
F(p) = 0 (mod p^3) are the Wolstenholme primes: 16843 and 2124679.
Odd primes p such that F(p-1) == 0 (mod p) are the Wieferich primes.
I also noticed that if p is an odd prime, then F(p+1) == 1 (mod p).
If p > 3 is a prime, then F(p^2) == 0 (mod p).
The series F(n) = A330718(n) / A330719(n) :
https://oeis.org/A330718 / https://oeis.org/A330719

The double sum of Fermat fractions,
S(n) = Sum_{k=1..n} F(k), where F(n) = Sum_{k=1..n} (2^{k-1}-1)/k as above;
0/1, 1/2, 2/1, 21/4, 23/2, 275/12, 130/3, 637/8, 577/4, 10399/40, 4687/10,
101761/120, ...
If p is an odd prime then S(p-1) == -1 (mod p) and S(p) == -1 (mod p), as I
stated.
And most importantly, if p > 3 is a prime, then S(p+1) == 0 (mod p^2) [sic].
Finally, an observation: S(m) = even / odd if and only if m = 2^k-1.

The product of Fermat fractions P(n) = Product_{k=2..n} (2^k-2)/k :
https://oeis.org/history/view?seq=A091669&v=53 (see my formula).

See also https://oeis.org/A279683 and https://oeis.org/A331343

Best regards,

Thomas
_______________________
https://en.wikipedia.org/wiki/Wolstenholme%27s_theorem