A003474 Generalized Euler phi function (for p=3).

1, 4, 18, 32, 160, 324, 1456, 2048, 13122, 25600, 117128, 209952, 913952, 2119936, 9447840, 13107200, 86093440, 172186884, 774840976, 1310720000, 6964002864, 13718968384, 62761410632, 88159684608, 557885504000, 835308258304, 5083731656658, 8988257288192, 45753584909920, 89261680665600, 411782264189296, 564050001920000

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

Comments

For n >= 2, a(n) is the number of n X n circulant invertible matrices over GF(3). - Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 22 2003

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• J. T. B. Beard Jr. and K. I. West, Factorization tables for x^n-1 over GF(q), Math. Comp., 28 (1974), 1167-1168.
• Gabriele Fici and Estéban Gabory, Generalized De Bruijn Words, Invertible Necklaces, and the Burrows-Wheeler Transform, arXiv:2502.12844 [math.CO], 2025. See Table 2 p. 11.

Crossrefs

Cf. A003473 (p=2), A192037 (p=5).

Programs

Mathematica

p = 3; numNormalp[n_] := Module[{r, i, pp}, pp = 1; Do[r = MultiplicativeOrder[p, d]; i = EulerPhi[d]/r; pp *= (1-1/p^r)^i, {d, Divisors[n]}]; Return[pp]]; numNormal[n_] := Module[{t, q, pp}, t=1; q=n; While[0==Mod[q, p], q /= p; t += 1]; pp = numNormalp[q]; pp *= p^n/n; Return[pp]]; a[n_] := If[n==1, 1, n*numNormal[n]]; Array[a, 40] (* Jean-François Alcover, Dec 10 2015, after Joerg Arndt *)

PARI

p=3; /* global */
num_normal_p(n)=
{
    my( r, i, pp );
    pp = 1;
    fordiv (n, d,
        r = znorder(Mod(p,d));
        i = eulerphi(d)/r;
        pp *= (1 - 1/p^r)^i;
    );
    return( pp );
}
num_normal(n)=
{
    my( t, q, pp );
    t = 1;  q = n;
    while ( 0==(q%p), q/=p; t+=1; );
    /* here: n==q*p^t */
    pp = num_normal_p(q);
    pp *= p^n/n;
    return( pp );
}
a(n)=if ( n==1, 1, n * num_normal(n) );
v=vector(66,n,a(n))
/* Joerg Arndt, Jul 03 2011 */

Extensions

Terms > 86093440 from Joerg Arndt, Jul 03 2011