cp's OEIS Frontend

A320092 Number of primitive (=aperiodic) 8-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 A320092 #9 Dec 11 2020 10:56:12
%S A320092 1,8,71,575,4670,37367,299510,2396150,19173302,153386927,1227128750,
%T A320092 9817030070,78536506805,628292058542,5026338565487,40210708557167,
%U A320092 321685685267822,2573485482143150,20587883991625133,164703071933262773,1317624576539847542
%N A320092 Number of primitive (=aperiodic) 8-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 A320092 Alois P. Heinz, <a href="/A320092/b320092.txt">Table of n, a(n) for n = 1..1108</a>
%F A320092 a(n) = Sum_{j=1..n} Sum_{d|j} 8^(d-1) * mu(j/d).
%F A320092 a(n) = A143327(n,8).
%F A320092 a(n) = Sum_{j=1..n} A143325(j,8).
%F A320092 a(n) = A143326(n,8) / 8.
%F A320092 G.f.: (1/(1 - x)) * Sum_{k>=1} mu(k) * x^k / (1 - 8*x^k). - _Ilya Gutkovskiy_, Dec 11 2020
%p A320092 b:= n-> add(`if`(d=n, 8^(n-1), -b(d)), d=numtheory[divisors](n)):
%p A320092 a:= proc(n) option remember; b(n)+`if`(n<2, 0, a(n-1)) end:
%p A320092 seq(a(n), n=1..30);
%o A320092 (PARI) a(n) = sum(j=1, n, sumdiv(j, d, 8^(d-1)*moebius(j/d))); \\ _Michel Marcus_, Dec 11 2020
%Y A320092 Column k=8 of A143327.
%Y A320092 Partial sums of A320073.
%Y A320092 Cf. A008683, A143325, A143326.
%K A320092 nonn
%O A320092 1,2
%A A320092 _Alois P. Heinz_, Oct 05 2018