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A003473 Generalized Euler phi function (for p=2).
(Formerly M0875)
8
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

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)

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-Fran├žois 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: A122412 A174019 A128035 * A095373 A249357 A056802

Adjacent sequences:  A003470 A003471 A003472 * A003474 A003475 A003476

KEYWORD

nonn

AUTHOR

N. J. A. Sloane.

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 August 23 19:19 EDT 2017. Contains 291021 sequences.