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A003473
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Generalized Euler phi function (for p=2).
(Formerly M0875)
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10
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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
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OFFSET
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1,2
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COMMENTS
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a(n) is the number of n X n circulant invertible matrices over GF(2). - Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 20 2003
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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Alois P. Heinz, Table of n, a(n) for n = 1..1000
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)
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FORMULA
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a(n) = n * A027362(n). - Vladeta Jovovic, Sep 09 2003
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MATHEMATICA
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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 *)
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PROG
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(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 */
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CROSSREFS
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Cf. A003474 (p=3), A192037 (p=5).
Cf. also A086479, A027362.
Sequence in context: A356371 A293389 A128035 * A095373 A249357 A291400
Adjacent sequences: A003470 A003471 A003472 * A003474 A003475 A003476
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KEYWORD
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nonn
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AUTHOR
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N. J. A. Sloane
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EXTENSIONS
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More terms from Vladeta Jovovic, Sep 09 2003
Terms > 331776000 from Joerg Arndt, Jul 03 2011
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STATUS
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approved
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