A320090 Number of primitive (=aperiodic) 6-ary words with length less than or equal to n which are earlier in lexicographic order than any other word derived by cyclic shifts of the alphabet.

1, 6, 41, 251, 1546, 9281, 55936, 335656, 2015236, 12091631, 72557806, 435346876, 2612129211, 15672776566, 94036939331, 564221643971, 3385331551426, 20311989308806, 121871945977221, 731231675909811, 4387390115926096, 26324340695837771, 157946044538104906

Offset

1,2

Links

Alois P. Heinz, Table of n, a(n) for n = 1..1285

Formula

a(n) = Sum_{j=1..n} Sum_{d|j} 6^(d-1) * mu(j/d).

a(n) = A143327(n,6).

a(n) = Sum_{j=1..n} A143325(j,6).

a(n) = A143326(n,6) / 6.

G.f.: (1/(1 - x)) * Sum_{k>=1} mu(k) * x^k / (1 - 6*x^k). - Ilya Gutkovskiy, Dec 11 2020

Maple

b:= n-> add(`if`(d=n, 6^(n-1), -b(d)), d=numtheory[divisors](n)):
a:= proc(n) option remember; b(n)+`if`(n<2, 0, a(n-1)) end:
seq(a(n), n=1..30);

Mathematica

nmax = 20; Rest[CoefficientList[Series[1/(1-x) * Sum[MoebiusMu[k] * x^k / (1 - 6*x^k), {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Dec 11 2020 *)

Prog

(PARI) a(n) = sum(j=1, n, sumdiv(j, d, 6^(d-1)*moebius(j/d))); \\ Michel Marcus, Dec 11 2020

Crossrefs

Column k=6 of A143327.

Partial sums of A320071.

Cf. A008683, A143325, A143326.

Sequence in context: A197196 A295117 A198795 * A266425 A194781 A323969

Adjacent sequences: A320087 A320088 A320089 * A320091 A320092 A320093

Keyword

nonn

Author

Alois P. Heinz, Oct 05 2018

Status

approved