cp's OEIS Frontend

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.

%I A320091 #9 Dec 11 2020 10:56:01
%S A320091 1,7,55,391,2791,19543,137191,960391,6725143,47076343,329551591,
%T A320091 2306861191,16148148391,113037041143,791260111543,5538820797943,
%U A320091 38771751367543,271402259573191,1899815857483639,13298711002502839,93090977299997143,651636841100805895
%N 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.
%H A320091 Alois P. Heinz, <a href="/A320091/b320091.txt">Table of n, a(n) for n = 1..1184</a>
%F A320091 a(n) = Sum_{j=1..n} Sum_{d|j} 7^(d-1) * mu(j/d).
%F A320091 a(n) = A143327(n,7).
%F A320091 a(n) = Sum_{j=1..n} A143325(j,7).
%F A320091 a(n) = A143326(n,7) / 7.
%F A320091 G.f.: (1/(1 - x)) * Sum_{k>=1} mu(k) * x^k / (1 - 7*x^k). - _Ilya Gutkovskiy_, Dec 11 2020
%p A320091 b:= n-> add(`if`(d=n, 7^(n-1), -b(d)), d=numtheory[divisors](n)):
%p A320091 a:= proc(n) option remember; b(n)+`if`(n<2, 0, a(n-1)) end:
%p A320091 seq(a(n), n=1..30);
%o A320091 (PARI) a(n) = sum(j=1, n, sumdiv(j, d, 7^(d-1)*moebius(j/d))); \\ _Michel Marcus_, Dec 11 2020
%Y A320091 Column k=7 of A143327.
%Y A320091 Partial sums of A320072.
%Y A320091 Cf. A008683, A143325, A143326.
%K A320091 nonn
%O A320091 1,2
%A A320091 _Alois P. Heinz_, Oct 05 2018