[seqfan] The fractions A330718(n) / A330719(n)

[Tomasz Ordowski]

Hello SeqFans!

Let N(n) = Numerator(Sum_{k=1..n} (2^k-2)/k).
0, 1, 3, 13, 25, 137, 245, 871, 517, 4629, 8349, ...
N(n) = Numerator(Sum_{k=1..n} (2^{k-1}-1)/k).
I noticed that if p > 3 is prime, then p^2 | N(p).
Note the similarity to Wolstenholme's theorem.
For n > 3, if n^2 | N(n), then n is prime, I think.
The weak pseudoprimes m such that m | N(m);
maybe someone will find such composites m.
Primes p such that p^3 | N(p) are A088164,
https://oeis.org/A088164 [proof needed].

Let D(n) = Denominator(Sum_{k=1..n} (2^{k-1}-1)/k).
1, 2, 2, 4, 4, 12, 12, 24, 8, 40, 40, 120, 120, 840, 840, ...
Conjecture: if p is an odd prime, then p | N(p+1) - D(p+1).
Is the composite number 25 the only pseudoprime here?
Primes p such that p^2 | N(p+1) - D(p+1) are 3, 5, 45827.

HAPPY NEW YEAR !

Thomas Ordowski
_______________________
Cf. https://oeis.org/A001008 / https://oeis.org/A002805
See https://oeis.org/A330718 / https://oeis.org/A330719
http://mathworld.wolfram.com/WolstenholmesTheorem.html
See also https://oeis.org/A064169 (the last comment). Thanks!