cp's OEIS Frontend

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.

%I A320093 #9 Dec 11 2020 10:56:21
%S A320093 1,9,89,809,7369,66329,597769,5380009,48426649,435840569,3922624969,
%T A320093 35303624809,317733161289,2859598458169,25736390906489,
%U A320093 231627518218169,2084647707070009,18761829363630889,168856464660630009,1519708181946200889,13677373641002598169
%N 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.
%H A320093 Alois P. Heinz, <a href="/A320093/b320093.txt">Table of n, a(n) for n = 1..1048</a>
%F A320093 a(n) = Sum_{j=1..n} Sum_{d|j} 9^(d-1) * mu(j/d).
%F A320093 a(n) = A143327(n,9).
%F A320093 a(n) = Sum_{j=1..n} A143325(j,9).
%F A320093 a(n) = A143326(n,9) / 9.
%F A320093 G.f.: (1/(1 - x)) * Sum_{k>=1} mu(k) * x^k / (1 - 9*x^k). - _Ilya Gutkovskiy_, Dec 11 2020
%p A320093 b:= n-> add(`if`(d=n, 9^(n-1), -b(d)), d=numtheory[divisors](n)):
%p A320093 a:= proc(n) option remember; b(n)+`if`(n<2, 0, a(n-1)) end:
%p A320093 seq(a(n), n=1..30);
%o A320093 (PARI) a(n) = sum(j=1, n, sumdiv(j, d, 9^(d-1)*moebius(j/d))); \\ _Michel Marcus_, Dec 11 2020
%Y A320093 Column k=9 of A143327.
%Y A320093 Partial sums of A320074.
%Y A320093 Cf. A008683, A143325, A143326.
%K A320093 nonn
%O A320093 1,2
%A A320093 _Alois P. Heinz_, Oct 05 2018