|
|
A331343
|
|
a(n) = lcm(1,2,...,n) * Sum_{k=1..n} (2^(k-1) - 1) / k.
|
|
0
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,3
|
|
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
|
|
|
FORMULA
|
|
|
MATHEMATICA
|
a[n_] := LCM @@ Range[n] * Sum[(2^(k-1) - 1) / k, {k, 1, n}]; Array[a, 25]
|
|
PROG
|
(Magma) [Lcm([1..n])*&+[(2^(k-1)-1)/k:k in [1..n]]:n in [1..25]]; // Marius A. Burtea, Jan 14 2020
(PARI) a(n) = lcm([1..n])*sum(k=1, n, (2^(k-1) - 1) / k); \\ Michel Marcus, Jan 14 2020
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|