cp's OEIS Frontend

A320089 Number of primitive (=aperiodic) 5-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 A320089 #11 Dec 11 2020 05:59:13
%S A320089 1,5,29,149,773,3869,19493,97493,488093,2440589,12206213,61031093,
%T A320089 305171717,1525859213,7629374189,38146874189,190734764813,
%U A320089 953673824213,4768371089837,23841855464717,119209287089693,596046435527189,2980232226542813,14901161132714813
%N A320089 Number of primitive (=aperiodic) 5-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 A320089 Alois P. Heinz, <a href="/A320089/b320089.txt">Table of n, a(n) for n = 1..1431</a>
%F A320089 a(n) = Sum_{j=1..n} Sum_{d|j} 5^(d-1) * mu(j/d).
%F A320089 a(n) = A143327(n,5).
%F A320089 a(n) = Sum_{j=1..n} A143325(j,5).
%F A320089 a(n) = A143326(n,5) / 5.
%F A320089 G.f.: (1/(1 - x)) * Sum_{k>=1} mu(k) * x^k / (1 - 5*x^k). - _Ilya Gutkovskiy_, Dec 11 2020
%p A320089 b:= n-> add(`if`(d=n, 5^(n-1), -b(d)), d=numtheory[divisors](n)):
%p A320089 a:= proc(n) option remember; b(n)+`if`(n<2, 0, a(n-1)) end:
%p A320089 seq(a(n), n=1..30);
%t A320089 nmax = 20; Rest[CoefficientList[Series[1/(1-x) * Sum[MoebiusMu[k] * x^k / (1 - 5*x^k), {k, 1, nmax}], {x, 0, nmax}], x]] (* _Vaclav Kotesovec_, Dec 11 2020 *)
%o A320089 (PARI) a(n) = sum(j=1, n, sumdiv(j, d, 5^(d-1)*moebius(j/d))); \\ _Michel Marcus_, Dec 11 2020
%Y A320089 Column k=5 of A143327.
%Y A320089 Partial sums of A295506.
%Y A320089 Cf. A008683, A143325, A143326.
%K A320089 nonn
%O A320089 1,2
%A A320089 _Alois P. Heinz_, Oct 05 2018