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A331343 a(n) = lcm(1,2,...,n) * Sum_{k=1..n} (2^(k-1) - 1) / k.

%I A331343 #11 Sep 08 2022 08:46:25
%S A331343 0,1,9,39,375,685,8575,30485,162855,291627,5785857,10514427,250200951,
%T A331343 461037291,854622483,3185234481,101381371377,190598779657,
%U A331343 6833215763803,12935721409039,24559552771039,46750514134519,2051664357879617,3923102768811707,37581323659852375
%N A331343 a(n) = lcm(1,2,...,n) * Sum_{k=1..n} (2^(k-1) - 1) / k.
%C A331343 By Wolstenholme's theorem, if p > 3 is a prime, then p^3 | a(p).
%C A331343 Conjecture: for n > 3, if n^3 | a(n), then n is prime. If so, there are no such pseudoprimes.
%C A331343 Problem: are there weak pseudoprimes m such that m^2 | a(m)? None up to 5*10^4.
%C A331343 Composite numbers m such that m | a(m) are 9, 25, 49, 99, 121, 125, 169, 221, 289, 343, 357, 361, 399, 529, 665, 841, 961, 1331, 1369, 1443, 1681, 1849, 2183, ...  Cf. A082180.
%C A331343 Prime numbers p such that p^4 | a(p) are probably only the Wolstenholme primes A088164.
%H A331343 Wikipedia, <a href="https://en.wikipedia.org/wiki/Wolstenholme%27s_theorem">Wolstenholme's theorem</a>.
%F A331343 a(n) = A003418(n) * A330718(n) / A330719(n).
%t A331343 a[n_] := LCM @@ Range[n] * Sum[(2^(k-1) - 1) / k, {k, 1, n}]; Array[a, 25]
%o A331343 (Magma) [Lcm([1..n])*&+[(2^(k-1)-1)/k:k in [1..n]]:n in [1..25]]; // _Marius A. Burtea_, Jan 14 2020
%o A331343 (PARI) a(n) = lcm([1..n])*sum(k=1, n, (2^(k-1) - 1) / k); \\ _Michel Marcus_, Jan 14 2020
%Y A331343 Cf. A003418, A025529, A082180, A088164, A330718, A330719.
%K A331343 nonn
%O A331343 1,3
%A A331343 _Amiram Eldar_ and _Thomas Ordowski_, Jan 14 2020