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

0, 1, 9, 39, 375, 685, 8575, 30485, 162855, 291627, 5785857, 10514427, 250200951, 461037291, 854622483, 3185234481, 101381371377, 190598779657, 6833215763803, 12935721409039, 24559552771039, 46750514134519, 2051664357879617, 3923102768811707, 37581323659852375

Offset: 1

Author

Amiram Eldar and Thomas Ordowski, Jan 14 2020

Keywords

• nonn

Comments

By Wolstenholme's theorem, if p > 3 is a prime, then p^3 | a(p).

Conjecture: for n > 3, if n^3 | a(n), then n is prime. If so, there are no such pseudoprimes.

Problem: are there weak pseudoprimes m such that m^2 | a(m)? None up to 5*10^4.

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.

Prime numbers p such that p^4 | a(p) are probably only the Wolstenholme primes A088164.

Links

• Wikipedia, Wolstenholme's theorem.

Crossrefs

Cf. A003418, A025529, A082180, A088164, A330718, A330719.

Programs

Magma

[Lcm([1..n])*&+[(2^(k-1)-1)/k:k in [1..n]]:n in [1..25]]; // Marius A. Burtea, Jan 14 2020

Mathematica

a[n_] := LCM @@ Range[n] * Sum[(2^(k-1) - 1) / k, {k, 1, n}]; Array[a, 25]

PARI

a(n) = lcm([1..n])*sum(k=1, n, (2^(k-1) - 1) / k); \\ Michel Marcus, Jan 14 2020

Formula

a(n) = A003418(n) * A330718(n) / A330719(n).