A320095 Number of primitive (=aperiodic) n-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, 2, 11, 79, 773, 9281, 137191, 2396150, 48426649, 1111099879, 28531150811, 810554312866, 25239591811405, 854769747700454, 31278135014945519, 1229782937960902111, 51702516367459973873, 2314494592652832016030, 109912203092221714132219, 5518821052631039996623577

Offset

1,2

Links

Alois P. Heinz, Table of n, a(n) for n = 1..387

Formula

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

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

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

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

a(n) = [x^n] (1/(1 - x)) * Sum_{k>=1} mu(k) * x^k / (1 - n*x^k). - Ilya Gutkovskiy, Feb 16 2020

Maple

b:= (n, k)-> add(`if`(d=n, k^(n-1), -b(d, k)), d=numtheory[divisors](n)):
g:= proc(n, k) option remember; b(n, k)+`if`(n<2, 0, g(n-1, k)) end:
a:= n-> g(n$2):
seq(a(n), n=1..23);

Mathematica

a[n_] := Sum[n^(d-1)*MoebiusMu[j/d], {j, 1, n}, {d, Divisors[j]}];
Table[a[n], {n, 1, 20}] (* Jean-François Alcover, Oct 25 2022, after A143327 *)

Prog

(PARI) a(n) = sum(j=1, n, sumdiv(j, d, n^(d-1) * moebius(j/d))); \\ Michel Marcus, Feb 16 2020

Crossrefs

Main diagonal of A143327.

Cf. A143325, A143326.

Sequence in context: A253256 A163203 A142722 * A099661 A027110 A341955

Adjacent sequences: A320092 A320093 A320094 * A320096 A320097 A320098

Keyword

nonn

Author

Alois P. Heinz, Oct 05 2018

Status

approved