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A003474 Generalized Euler phi function (for p=3).
(Formerly M3541)
4

%I M3541 #34 Oct 21 2023 19:20:42

%S 1,4,18,32,160,324,1456,2048,13122,25600,117128,209952,913952,2119936,

%T 9447840,13107200,86093440,172186884,774840976,1310720000,6964002864,

%U 13718968384,62761410632,88159684608,557885504000,835308258304,5083731656658,8988257288192,45753584909920,89261680665600,411782264189296,564050001920000

%N Generalized Euler phi function (for p=3).

%C 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

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

%H J. T. B. Beard Jr. and K. I. West, <a href="http://dx.doi.org/10.1090/S0025-5718-1974-0364196-5">Factorization tables for x^n-1 over GF(q)</a>, Math. Comp., 28 (1974), 1167-1168.

%t 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_ *)

%o (PARI)

%o p=3; /* global */

%o num_normal_p(n)=

%o {

%o my( r, i, pp );

%o pp = 1;

%o fordiv (n, d,

%o r = znorder(Mod(p,d));

%o i = eulerphi(d)/r;

%o pp *= (1 - 1/p^r)^i;

%o );

%o return( pp );

%o }

%o num_normal(n)=

%o {

%o my( t, q, pp );

%o t = 1; q = n;

%o while ( 0==(q%p), q/=p; t+=1; );

%o /* here: n==q*p^t */

%o pp = num_normal_p(q);

%o pp *= p^n/n;

%o return( pp );

%o }

%o a(n)=if ( n==1, 1, n * num_normal(n) );

%o v=vector(66,n,a(n))

%o /* _Joerg Arndt_, Jul 03 2011 */

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

%K nonn

%O 1,2

%A _N. J. A. Sloane_

%E Terms > 86093440 from _Joerg Arndt_, Jul 03 2011

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Last modified March 28 05:02 EDT 2024. Contains 371235 sequences. (Running on oeis4.)