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
Keywords
Links
- Alois P. Heinz, Table of n, a(n) for n = 1..387
Programs
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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);
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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 *) -
PARI
a(n) = sum(j=1, n, sumdiv(j, d, n^(d-1) * moebius(j/d))); \\ Michel Marcus, Feb 16 2020
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