A320093 Number of primitive (=aperiodic) 9-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, 9, 89, 809, 7369, 66329, 597769, 5380009, 48426649, 435840569, 3922624969, 35303624809, 317733161289, 2859598458169, 25736390906489, 231627518218169, 2084647707070009, 18761829363630889, 168856464660630009, 1519708181946200889, 13677373641002598169

Offset

1,2

Links

Alois P. Heinz, Table of n, a(n) for n = 1..1048

Formula

a(n) = Sum_{j=1..n} Sum_{d|j} 9^(d-1) * mu(j/d).

a(n) = A143327(n,9).

a(n) = Sum_{j=1..n} A143325(j,9).

a(n) = A143326(n,9) / 9.

G.f.: (1/(1 - x)) * Sum_{k>=1} mu(k) * x^k / (1 - 9*x^k). - Ilya Gutkovskiy, Dec 11 2020

Maple

b:= n-> add(`if`(d=n, 9^(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);

Prog

(PARI) a(n) = sum(j=1, n, sumdiv(j, d, 9^(d-1)*moebius(j/d))); \\ Michel Marcus, Dec 11 2020

Crossrefs

Column k=9 of A143327.

Partial sums of A320074.

Cf. A008683, A143325, A143326.

Sequence in context: A133486 A224760 A198967 * A015584 A072256 A138288

Adjacent sequences: A320090 A320091 A320092 * A320094 A320095 A320096

Keyword

nonn

Author

Alois P. Heinz, Oct 05 2018

Status

approved