A094678 a(n) = A003474(n)/n.

1, 2, 6, 8, 32, 54, 208, 256, 1458, 2560, 10648, 17496, 70304, 151424, 629856, 819200, 5064320, 9565938, 40781104, 65536000, 331619184, 623589472, 2728756984, 3673320192, 22315420160, 32127240704, 188286357654, 321009188864, 1577709824480, 2975389355520, 13283298844816, 17626562560000

Offset: 1

Author

Vladeta Jovovic, Jun 07 2004

Keywords

• nonn

Comments

Number of normal bases for GF(3^n) over GF(3). - Joerg Arndt, Jul 03 2011

For n>=2, a(n) = f(n)/(2^(n-1)) where f(n) is the number of Hamiltonian cycles in the 3-ary de Bruijn graph (i.e., graph with 3*n nodes {0..3*n-1} and edges from each i to 3*i (mod 3*n), 3*i+1 (mod 3*n), and 3*i+2 (mod 3*n); cf. A192513). - Joerg Arndt, Jul 03 2011.

For details on this correspondence, see A192513. - Dmitrii Pasechnik, Dec 07 2014

Programs

Mathematica

p = 3; numNormalp[n_] := Module[{r, i, pp = 1}, Do[r = MultiplicativeOrder[p, d]; i = EulerPhi[d]/r; pp *= (1 - 1/p^r)^i, {d, Divisors[n]}]; Return[pp]];
a[1] = 1; a[n_] := Module[{t = 1, q = n, pp}, While[0 == Mod[q, p], q /= p; t += 1]; pp = numNormalp[q]; pp *= p^n/n; Return[pp]];
Array[a, 40] (* Jean-François Alcover, Jul 22 2018, after Joerg Arndt *)

PARI

a(n)=if(n==1,return(1));my(r,i,t=3^n/n);fordiv(n/3^valuation(n,3), d, r=znorder(Mod(3,d)); i=eulerphi(d)/r; t*=(1-1/3^r)^i);t \\ Charles R Greathouse IV, Jan 03 2013

Extensions

Terms > 5064320 by Joerg Arndt, Jul 03 2011