A320088 Number of primitive (=aperiodic) 4-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, 4, 19, 79, 334, 1339, 5434, 21754, 87274, 349159, 1397734, 5590954, 22368169, 89472934, 357908119, 1431633559, 5726600854, 22906403494, 91625880229, 366503524969, 1466015148634, 5864060611159, 23456246655574, 93824986622614, 375299963333014, 1501199853398419

Offset: 1

Author

Alois P. Heinz, Oct 05 2018

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1661

Crossrefs

Column k=4 of A143327.

Partial sums of A295505.

Cf. A008683, A143325, A143326.

Programs

Maple

b:= n-> add(`if`(d=n, 4^(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 - 4*x^k), {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Dec 11 2020 *)

PARI

a(n) = sum(j=1, n, sumdiv(j, d, 4^(d-1)*moebius(j/d))); \\ Michel Marcus, Dec 11 2020

Formula

a(n) = Sum_{j=1..n} Sum_{d|j} 4^(d-1) * mu(j/d).

a(n) = A143327(n,4).

a(n) = Sum_{j=1..n} A143325(j,4).

a(n) = A143326(n,4) / 4.

G.f.: (1/(1 - x)) * Sum_{k>=1} mu(k) * x^k / (1 - 4*x^k). - Ilya Gutkovskiy, Dec 11 2020