A320091 Number of primitive (=aperiodic) 7-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, 7, 55, 391, 2791, 19543, 137191, 960391, 6725143, 47076343, 329551591, 2306861191, 16148148391, 113037041143, 791260111543, 5538820797943, 38771751367543, 271402259573191, 1899815857483639, 13298711002502839, 93090977299997143, 651636841100805895

Offset: 1

Author

Alois P. Heinz, Oct 05 2018

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1184

Crossrefs

Column k=7 of A143327.

Partial sums of A320072.

Cf. A008683, A143325, A143326.

Programs

Maple

b:= n-> add(`if`(d=n, 7^(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);

PARI

a(n) = sum(j=1, n, sumdiv(j, d, 7^(d-1)*moebius(j/d))); \\ Michel Marcus, Dec 11 2020

Formula

a(n) = Sum_{j=1..n} Sum_{d|j} 7^(d-1) * mu(j/d).

a(n) = A143327(n,7).

a(n) = Sum_{j=1..n} A143325(j,7).

a(n) = A143326(n,7) / 7.

G.f.: (1/(1 - x)) * Sum_{k>=1} mu(k) * x^k / (1 - 7*x^k). - Ilya Gutkovskiy, Dec 11 2020