[seqfan] Nice divisibility

[Tomasz Ordowski]

Dear readers!

Let a(n) = Numerator(-1/n + Sum_{k=1..n} 2^{k-1}/k).
0, 3, 3, 61, 25, 137, 343, 32663, 2357, 74689, 66671, ...
I noticed that if p > 3 is prime, then p^2 | a(p). Nice!
   Does it follow from Wolstenholme's theorem?
If p is a Wolstenholme prime, then p^3 | a(p).
No composites m such that m^2 | a(m).
Note that 7^2 | a(7^2) and 7^3 | a(7).
Thanks to Amiram Eldar.

Best regards,

Thomas Ordowski
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Similarly: b(n) = Numerator(-2/n + Sum_{k=1..n} 2^k/k).
0, 3, 6, 61, 50, 137, 686, 32663, 4714, 74689, 133342, ...
Also here we have the divisibility p^2 | b(p) for primes p > 3.
http://mathworld.wolfram.com/WolstenholmesTheorem.html
https://en.wikipedia.org/wiki/Wolstenholme%27s_theorem
Cf. https://oeis.org/A330718 and https://oeis.org/A001008
Cf. https://oeis.org/A108866 (see the reference).
Does anyone know p-adic analysis?