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

Author

Alois P. Heinz, Oct 05 2018

Keywords

• nonn

Links

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

Crossrefs

Column k=6 of A143327.

Partial sums of A320071.

Cf. A008683, A143325, A143326.

Programs

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 *)

PARI

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

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