



1, 2, 2, 4, 2, 4, 2, 4, 2, 4, 2, 8, 2, 4, 4, 4, 2, 4, 2, 8, 4, 4, 2, 8, 2, 4, 2, 8, 2, 8, 2, 4, 4, 4, 4, 8, 2, 4, 4, 8, 2, 8, 2, 8, 4, 4, 2, 8, 2, 4, 4, 8, 2, 4, 4, 8, 4, 4, 2, 16, 2, 4, 4, 4, 4, 8, 2, 8, 4, 8, 2, 8, 2, 4, 4, 8, 4, 8, 2, 8, 2, 4, 2, 16, 4, 4, 4, 8, 2, 8, 4, 8, 4, 4, 4, 8, 2, 4, 4, 8
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OFFSET

0,2


COMMENTS

a(n) is the number of quadratic number fields Q[sqrt(d)] (including Q itself) that are subfields of the cyclotomic field Q[exp(Pi*i/n)], where i is the imaginary unit. Note that for odd k, Q[exp(2*Pi*i/k)] = Q[exp(2*Pi*i/(2*k))], so we can just consider the case Q[exp(2*Pi*i/(2*k))] for integers k and let n = 2*k.
a(n) = 2 if and only if n = 2 or n = p^e, where p is an odd prime and e >= 1.


LINKS

Table of n, a(n) for n=0..99.


FORMULA

a(n) = 2*A060594(n) if n is even and not divisible by 8, otherwise A060594(n).
Multiplicative with a(2) = 2 and a(2^e) = 4 for e > 1; a(p^e) = 2 for odd primes p.
a(n) = 2^omega(n) if 4 does not divide n, otherwise 2^(omega(n)+1), omega = A001221.


EXAMPLE

List of quadratic number fields (including Q itself) that are subfields of Q[exp(Pi*i/n)]:
n = 2 (the quotient field over the Gaussian integers): Q, Q[i];
n = 3 (the quotient field over the Eisenstein integers): Q, Q[sqrt(3)];
n = 4: Q, Q[sqrt(2)], Q[i], Q[sqrt(2)];
n = 5: Q, Q[sqrt(5)];
n = 6: Q, Q[sqrt(3)], Q[sqrt(3)], Q[i];
n = 7: Q, Q[sqrt(7)];
n = 8: Q, Q[sqrt(2)], Q[i], Q[sqrt(2)];
n = 9: Q, Q[sqrt(3)];
n = 10: Q, Q[sqrt(5)], Q[i], Q[sqrt(5)];
n = 11: Q, Q[sqrt(11)];
n = 12: Q, Q[sqrt(2)], Q[sqrt(3)], Q[sqrt(6)], Q[sqrt(3)], Q[i], Q[sqrt(2)], Q[sqrt(6)];
n = 13: Q, Q[sqrt(13)];
n = 14: Q, Q[sqrt(7)], Q[i], Q[sqrt(7)];
n = 15: Q, Q[sqrt(5)], Q[sqrt(3)], Q[sqrt(15)];
n = 16: Q, Q[sqrt(2)], Q[i], Q[sqrt(2)].


PROG

(PARI) a(n) = 2^#znstar(2*n)[2]


CROSSREFS

Cf. A060594, A001221.
Sequence in context: A040003 A106469 A082508 * A303809 A193562 A249868
Adjacent sequences: A327727 A327728 A327729 * A327731 A327732 A327733


KEYWORD

nonn,mult


AUTHOR

Jianing Song, Sep 23 2019


STATUS

approved



