A252764 Number of length n primitive (=aperiodic or period n) n-ary words.

1, 2, 24, 240, 3120, 46410, 823536, 16773120, 387419760, 9999899910, 285311670600, 8916097441680, 302875106592240, 11112006720144330, 437893890380096640, 18446744069414584320, 827240261886336764160, 39346408075098144278664, 1978419655660313589123960

Offset: 1

Author

Alois P. Heinz, Dec 21 2014

Keywords

• nonn

Examples

			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.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..350

Crossrefs

Main diagonal of A143324.

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

Programs

Maple

with(numtheory):
a:= n-> add(n^d *mobius(n/d), d=divisors(n)):
seq(a(n), n=1..25);

Mathematica

a[n_] := DivisorSum[n, n^# * MoebiusMu[n/#]& ];
Array[a, 25] (* Jean-François Alcover, Mar 24 2017, translated from Maple *)

Formula

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

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

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

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

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