A298941 Number of permutations of the multiset of prime factors of n > 1 that are Lyndon words.

1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 2, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 0, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 0, 1, 1, 3, 1, 1, 1, 1, 1, 3

Offset

2,29

Links

Alois P. Heinz, Table of n, a(n) for n = 2..20000

Example

The a(90) = 3 Lyndon permutations are {2,3,3,5}, {2,3,5,3}, {2,5,3,3}.

Maple

with(combinat): with(numtheory):
g:= l-> (n-> `if`(n=0, 1, add(mobius(j)*multinomial(n/j,
        (l/j)[]), j=divisors(igcd(l[])))/n))(add(i, i=l)):
a:= n-> g(map(i-> i[2], ifactors(n)[2])):
seq(a(n), n=2..150);  # Alois P. Heinz, Feb 09 2018

Mathematica

primeMS[n_]:=If[n===1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
LyndonQ[q_]:=Array[OrderedQ[{q, RotateRight[q, #]}]&, Length[q]-1, 1, And]&&Array[RotateRight[q, #]&, Length[q], 1, UnsameQ];
Table[Length[Select[Permutations[primeMS[n]], LyndonQ]], {n, 2, 60}]
(* Second program: *)
multinomial[n_, k_List] := n!/Times @@ (k!);
g[l_] := With[{n = Total[l]}, If[n == 0, 1, Sum[MoebiusMu[j] multinomial[ n/j, l/j], {j, Divisors[GCD @@ l]}]/n]];
a[n_] := g[FactorInteger[n][[All, 2]]];
a /@ Range[2, 150] (* Jean-François Alcover, Dec 15 2020, after Alois P. Heinz *)

Crossrefs

Cf. A000740, A001037, A001222, A008480, A008965, A059966, A060223, A096443, A112798, A215366, A275024, A294859, A298947.

Sequence in context: A081396 A194293 A349595 * A317146 A194297 A100544

Adjacent sequences: A298938 A298939 A298940 * A298942 A298943 A298944

Keyword

nonn

Author

Gus Wiseman, Jan 29 2018

Status

approved