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A003473 Generalized Euler phi function (for p=2).
(Formerly M0875)
10
1, 2, 3, 8, 15, 24, 49, 128, 189, 480, 1023, 1536, 4095, 6272, 10125, 32768, 65025, 96768, 262143, 491520, 583443, 2095104, 4190209, 6291456, 15728625, 33546240, 49545027, 102760448, 268435455, 331776000, 887503681, 2147483648, 3211797501, 8522956800, 12325233375, 25367150592, 68719476735, 137438429184, 206007472125 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
a(n) is the number of n X n circulant invertible matrices over GF(2). - Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 20 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.
Swee Hong Chan, Henk D. L. Hollmann, Dmitrii V. Pasechnik, Sandpile groups of generalized de Bruijn and Kautz graphs and circulant matrices over finite fields, arXiv:1405.0113 [math.CO], (1-May-2014)
FORMULA
a(n) = n * A027362(n). - Vladeta Jovovic, Sep 09 2003
MATHEMATICA
p = 2; 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_] := n*numNormal[n]; Array[a, 40] (* Jean-François Alcover, Dec 10 2015, after Joerg Arndt *)
PROG
(PARI)
p=2; /* 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)=n * num_normal(n);
v=vector(66, n, a(n)) /* Joerg Arndt, Jul 03 2011 */
CROSSREFS
Cf. A003474 (p=3), A192037 (p=5).
Cf. also A086479, A027362.
Sequence in context: A356371 A293389 A128035 * A095373 A249357 A291400
KEYWORD
nonn
AUTHOR
EXTENSIONS
More terms from Vladeta Jovovic, Sep 09 2003
Terms > 331776000 from Joerg Arndt, Jul 03 2011
STATUS
approved

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Last modified May 23 21:14 EDT 2024. Contains 372765 sequences. (Running on oeis4.)