A252764 - OEIS

%I #12 Jan 25 2018 11:49:37

%S 1,2,24,240,3120,46410,823536,16773120,387419760,9999899910,

%T 285311670600,8916097441680,302875106592240,11112006720144330,

%U 437893890380096640,18446744069414584320,827240261886336764160,39346408075098144278664,1978419655660313589123960

%N Number of length n primitive (=aperiodic or period n) n-ary words.

%H Alois P. Heinz, <a href="/A252764/b252764.txt">Table of n, a(n) for n = 1..350</a>

%F a(n) = Sum_{d|n} n^d * mu(n/d), mu = A008683.

%F a(n) = A075147(n)*n.

%F a(n) = A074650(n,n) * n.

%F a(n) = A143325(n,n) * n.

%F a(n) = A143324(n,n).

%e a(3) = 24 because there are 24 primitive words of length 3 over 3-letter alphabet {a,b,c}: aab, aac, aba, abb, abc, aca, acb, acc, baa, bab, bac, bba, bbc, bca, bcb, bcc, caa, cab, cac, cba, cbb, cbc, cca, ccb.

%p with(numtheory):

%p a:= n-> add(n^d *mobius(n/d), d=divisors(n)):

%p seq(a(n), n=1..25);

%t a[n_] := DivisorSum[n, n^# * MoebiusMu[n/#]& ];

%t Array[a, 25] (* _Jean-François Alcover_, Mar 24 2017, translated from Maple *)

%Y Main diagonal of A143324.

%Y Cf. A008683, A074650, A075147, A143324, A143325.

%K nonn

%O 1,2

%A _Alois P. Heinz_, Dec 21 2014