A320089 Number of primitive (=aperiodic) 5-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, 5, 29, 149, 773, 3869, 19493, 97493, 488093, 2440589, 12206213, 61031093, 305171717, 1525859213, 7629374189, 38146874189, 190734764813, 953673824213, 4768371089837, 23841855464717, 119209287089693, 596046435527189, 2980232226542813, 14901161132714813

Offset

1,2

Links

Alois P. Heinz, Table of n, a(n) for n = 1..1431

Formula

a(n) = Sum_{j=1..n} Sum_{d|j} 5^(d-1) * mu(j/d).

a(n) = A143327(n,5).

a(n) = Sum_{j=1..n} A143325(j,5).

a(n) = A143326(n,5) / 5.

G.f.: (1/(1 - x)) * Sum_{k>=1} mu(k) * x^k / (1 - 5*x^k). - Ilya Gutkovskiy, Dec 11 2020

Maple

b:= n-> add(`if`(d=n, 5^(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 - 5*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, 5^(d-1)*moebius(j/d))); \\ Michel Marcus, Dec 11 2020

Crossrefs

Column k=5 of A143327.

Partial sums of A295506.

Cf. A008683, A143325, A143326.

Sequence in context: A198764 A065541 A060926 * A239715 A260752 A098780

Adjacent sequences: A320086 A320087 A320088 * A320090 A320091 A320092

Keyword

nonn

Author

Alois P. Heinz, Oct 05 2018

Status

approved