A320090 - OEIS

%I #13 Dec 11 2020 06:58:57

%S 1,6,41,251,1546,9281,55936,335656,2015236,12091631,72557806,

%T 435346876,2612129211,15672776566,94036939331,564221643971,

%U 3385331551426,20311989308806,121871945977221,731231675909811,4387390115926096,26324340695837771,157946044538104906

%N Number of primitive (=aperiodic) 6-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 Alois P. Heinz, <a href="/A320090/b320090.txt">Table of n, a(n) for n = 1..1285</a>

%F a(n) = Sum_{j=1..n} Sum_{d|j} 6^(d-1) * mu(j/d).

%F a(n) = A143327(n,6).

%F a(n) = Sum_{j=1..n} A143325(j,6).

%F a(n) = A143326(n,6) / 6.

%F G.f.: (1/(1 - x)) * Sum_{k>=1} mu(k) * x^k / (1 - 6*x^k). - _Ilya Gutkovskiy_, Dec 11 2020

%p b:= n-> add(`if`(d=n, 6^(n-1), -b(d)), d=numtheory[divisors](n)):

%p a:= proc(n) option remember; b(n)+`if`(n<2, 0, a(n-1)) end:

%p seq(a(n), n=1..30);

%t nmax = 20; Rest[CoefficientList[Series[1/(1-x) * Sum[MoebiusMu[k] * x^k / (1 - 6*x^k), {k, 1, nmax}], {x, 0, nmax}], x]] (* _Vaclav Kotesovec_, Dec 11 2020 *)

%o (PARI) a(n) = sum(j=1, n, sumdiv(j, d, 6^(d-1)*moebius(j/d))); \\ _Michel Marcus_, Dec 11 2020

%Y Column k=6 of A143327.

%Y Partial sums of A320071.

%Y Cf. A008683, A143325, A143326.

%K nonn

%O 1,2

%A _Alois P. Heinz_, Oct 05 2018