# [seqfan] Harmonic properties of the Fermat fraction series

Tomasz Ordowski tomaszordowski at gmail.com
Mon Jan 6 15:24:11 CET 2020

```Dear readers,

I defined the series of Fermat fractions, F(n) = Sum _{k=1..n}
(2^{k-1}-1)/k:
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 I noticed that if p > 3 is a prime, then F(p) == 0 (mod p^2) [sic].
Does this result from Wolstenholme's theorem? That is the question!
A positive answer is supported by the confirmed fact that primes p such
that
F(p) = 0 (mod p^3) are the Wolstenholme primes: 16843 and 2124679.
Note that if p is an odd prime, then F(p+1) == 1 (mod p).

I do not see generalizations to other bases.

Best regards,

Thomas Ordowski

P.S. Double sum of the Fermat fractions.
Let 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, 10339/40, 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].

```