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